hello everyone I welcome you all onto this Channel and my name is vinay Kumar from the gate mechanical team of PW and this video series is basically a single shot video series for mathematics of course and in this video we are going to take in a single shot for the complete linear algebra which is one very important module for gate mathematics and also for it actually consists of some two to three marks every year so we straight away start to this course linear algebra linear Al Jiva which many of you might be familiar with some different name which is also called matasis so many of you might have studied a bit of this with the name of this subject as or the name of this particular module as linear algebra now this medium of teaching will be in English and now it is you know English is basically the language of the education and 80 of the books across the world are written in English and I am telling you one thing even if you secure a good admission in some iits or iisc Bangalore in India at least you will experience all of your purposes almost will teach you in English so I uh you know I suggest you basically to have the understanding of the things things in English and definitely this will definitely you know help you out okay so straight away going to this chapter where you use this linear algebra if you ask this question to yourself many instances you will be solving simultaneous equations correct let us suppose if I give you this equation like a 1 1 X plus b one one y is equal to C1 and a two One X plus b 2 1 Y is equal to C2 for example let us suppose if I ask you to solve these two equations it's very easy simple for you right you have lot of techniques for you like elimination substitution you have lot of techniques but let us suppose if I try to increase the complexity for example instead of giving this 2D system let us suppose I have given you system like this a 1 2 X or you can say y plus a 1 3 Z is equal to C1 a to 1 X plus a two two y plus a 2 3 Z is equal to C2 and a the One X plus a Theta y plus a 32 32 you can say Z is equal to C3 so let us suppose if I give you this particular system it's at the equations with the unknowns then you know you can still solve this system using such kind of techniques but it's not very simple correct now let us suppose if you get four by four then it's much difficult case so basically matrices is the main application of this is you can solve simultaneous systems and the N number of applications in technical subjects for example many of you might have seen this Matrix like I was talking Tau XY Tau y x and sigma y this is basically a 2d test answer if you see instant of middle soft fluid mechanics you get such systems similarly if you are dealing with finite element analysis you will be getting stiffness matrices all these things okay so it has large number of applications and linear algebra is definitely one very good subject to deal with all such things in uh mathematics and myself when I was doing some research earlier I have dealt with a system of one not one unknowns and one or one equations also because you see generally many things change with respect to time location temperature there are certain things so I was able to solve one not one by one not one system using matrices again and of course with the help of some coding but right now my aim is to tell you that list linear algebra is one very important tool to crack all the you know such simultaneous systems in engineering okay so let us right away start from this Basics and this video will definitely help you out in understanding from the definition of thematics then slowly progressing to the advanced levels such that you can clear your all the exams basically and also a very good knowledge in matrices is clear so this video is helpful even for some je aspects as well if you try to go for the first two hours of this video it's a single shot video on this complete linear algebra and I'm very privileged to give you this information in a quick go before I you know gate examination okay so now let us try to start with the definition of a matrix so what is a matrix what is a matrix for example for example if you see many of you feel that any rectangular area of elements is called a matrix character in some books you find this definition a rectangular arrangement of elements in those and columns is called a matrix but you have seen Square matrices also then Square Matrix is not a rectangular Arrangement it's basically a square Arrangement so here I generally Define this as any arrangement uh you can write any array or you can say arrangement of elements of elements in horizontal lines in horizontal lines and you call this horizontal lines as a rows and vertical lines you call this vertical lines basically as columns is called is called a matrix is called a matrix now for example if I I'll definitely give one examples for each of the cases what we discussed so that it will you know make you better in understanding the things so examples if I take for example I can take normally this matrices generally you can write generally matrices are defined by or denoted by you can say generally matrices are denoted by are denoted by uppercase letters okay are denoted by uppercase letters means basically capital letters okay so uppercase alphabets you can say General uppercase alphabets okay so for example if I say a is equal to if I write these elements India you can write then uh what is the other five letter country okay you can say Japan for example so if you see this this Matrix then here all these elements i n d i a again j a p a and all these elements are arranged in some horizontal and also in vertical laws so basically this row is called the first and this o is called the second row and similarly if you see this column is called the first column similarly this column is the second column second column this is the third similarly followed by this fourth and finally we have this fifth column so this is your fifth column so if you see this example this Matrix capital A has two rows and five columns in it so what we say there is something called size or order of the Matrix okay so next we are going to for that definition size or order of The Matrix size or order of The Matrix now basically if you want to see the size of a matrix then the size is defined by number of horizontal laws you have and also number of vertical laws I mean basically the vertical lines basically the columns which you have okay so if a matrix has if a matrix capital A has M Rose and n columns n columns then it is denoted by it is denoted by a m by n so basically this is read as a m by n and if you see this m is basically number of rows that you have and this n is basically number of columns that you have here okay now how we Define matrices let us suppose I write a matrix for example okay so General notation of Matrix for example if I write this Matrix a is equal to 2 3 4 for example three four five and four five six for example let us Suppose there is some Matrix like this now many times in Gate examinations or in general in some standard textbooks or Advanced textbooks you find the notation of this Matrix in a different way means they tell about the same Matrix but in a different sense how look I'll tell you basically what they give you is for example if you take this Matrix this Matrix is equivalent as a is equal to a i j for all one less than or equal to I less than or equal to 3 1 less than or equal to J less than or equal to 3 and a i j is equal to I plus J so what is the meaning of this sentence how this Matrix and this Matrix both are same basically here so if you see normally if they denote a i j this first subscript means basically I denotes a row number okay so this I denotes row number and J denotes this J denotes column number okay so this J denotes column number so once if you want to specify location of any element in a matrix let us suppose if I want to specify this element then you know this element is in the second row and it's also in the second column so basically if I denotes the row number and J denotes the column number then I can say this element 4 which I have highlighted here is nothing but a 2 2 which means it belongs to the second or second column element okay so similarly if I want to spot about this four for example here then this 4 is in the third row and First Column so we call this as a one okay similarly any other for example if you take this 5 this is in the second or third column so we take as a two three and this element is given by some function of I comma J in general okay it can be I plus j i into J whatever any algebraic expression they can put sine I whatever they want they can put but if they want to generate a matrix my intention is to tell you that they need not completely specify the Matrix in this form but whether they can specify you in this form and here if you see I value is less than 1 I mean I is greater than 1 and less than 3 so basically if you see I can take the values of 1 2 and 3 which means this Matrix a which you are talking about has three O's and at the same time one less than or equal to J less than or equal to 3 that means this Matrix can have three columns so basically it's a three mathematics how do I know because row numbers are from 1 to 3 and also column numbers are from one to three and how do I know what are the elements inside this Matrix that is given by this for example if I want the element a one one then a i j is given as I plus J so I value is one J value is also one so this one plus one is equal to two similarly and if you see the first element is 2 in the first row for example if you want this element it's a third or second column element so third row second column element so this is three plus two and this value has to be five so if you see this element is 5 actually here so that's how you actually need not completely specify the Matrix Okay so so this is one very important way which you know they ask you questions in examinations and they actually Define this size with the help of this inequalities and they Define the elements using some function in terms of I and J so that's how this is one very important notation of matrices a clear and because I'm going to use this from no one words so that's why I'm you know going basically now I would like to give you some basic types of matrices because these types of mat assessments there are certain types of matrices like upper triangular lower triangular diagonal matrix all these things and these type of matrices they have certain special properties in determinants in inverse in eigen values all these things okay so we will see slowly in this video how we progress okay so types of matrices I'll quickly put a heading types of matrices types of matrices now the first type of Matrix that I am going to give you is upper triangular Matrix upper triangular upper triangular Matrix upper triangular Matrix so what is this upper triangular Matrix so basically if you take any Square Matrix then these elements for which I value is equal to J so 2 4 6 because if you take this first element this is a 1 1 then this element is a22 this element is a33 so basically if you take this diagonal for example let me show using a dotted line if you take this diagonal then all the elements on this diagonal have same number of I comma J so that's why we say since it looks like a diagonal we say those elements are called principal diagonal elements okay so this is the main diagonal so principal diagonal elements here clear so if you see this upper triangular Matrix a matrix ematics a matrix a is equal to a i j where one less than or equal to I comma J less than or equal to n this is how Channel you can write for square matrices you can denote both I and Jane a single inequality normally in previous case I have shown two different inequalities like one less than or equal to I less than or equal to 3 then 1 less than or equal to J less than or equal to 3 but it need not be like that once it's a square Matrix you can directly denote it okay so this Matrix is said to be is said to be n upper triangular Matrix is said to be n upper triangular Matrix is said to be an upper triangular Matrix if a i j is equal to 0 for all I greater than J so what is the meaning of this equation okay I have written this element will become 0 whenever I is greater than equal is greater than J basically here now what does this equation tells you this tells you that all the lower diagonal elements all the lower diagonal elements azios all the wide diagonal elements are zeros now how do you know this look for example let us suppose if I take this Matrix let us suppose if I take this Matrix a three by three Matrix let us take a 1 1 a 1 2 a 1 3 similarly a two one a two two a two three and a31 a32 and for example okay so now if you take this matrix it's a mathematics now I would like to tell something here look if you take this principle diagonal again if you take this principle diagonal now you can observe one thing if you take the elements Above This principle diagonal means basically the elements in this block if you see the elements in this block for example okay so if you see the elements in this block elements in this block you can say for example so if you take the elements in this block then you can observe one thing then you can observe one thing here that this I number will be greater than J number definitely for all the elements okay so all these elements which are below this principle diagonal are called the lower diagonal elements and all the elements which are above this principle diagonal are called the upper diagonal elements okay and now if you check this if you see the lower diagonal elements high number will be more than this J number for all the elements similarly if you take the upper diagonal I will be less than this J for all the elements so whenever I am telling I is greater than J it talks about these elements here okay so these are the elements the elements which are below this diagonal are the elements for which I value is actually greater than J so if you want to call any Matrix upper triangular you know all your lower diagonal elements must be Zeos which means this element should be 0 this element should be 0 and also this element should be 0 and I'm I don't bother about whether these elements being doesn't matter but as long as all these lower diagonal elements are zeros I call my Matrix as an upper triangular Matrix I hope it's clear to you so basically if you take any element below the principle diagonal you have your row number greater than the column number and for the Matrix to be called as an upper triangular Matrix all these lower diagonal elements should be definitely zero so clear so now with this understanding let us move to the second types of Matrix separate second type of Matrix which is lower triangular Matrix lower triangular Matrix Now by this time you might have definitely guessed what is the case Okay lower triangular Matrix lower triangular Matrix now what is the lower triangular Matrix look a matrix ematics a is equal to a i j where 1 less than or equal to I comma J less than or equal to n is said to be is said to be is said to be lower triangular is said to be lower triangular Matrix lower triangular Matrix if a i j is equal to 0 now you all can guess this I am pausing here for a second so you can write you can give the condition so way if a i j is equal to where I is less than J very good so a i is less than J so what is the meaning of this line so if you see all the upper diagonal elements are zeros all the upper diagonal elements all the upper diagonal elements elements are zeros all the upper diagonal elements are zeros okay so now if you see this condition example again as I have given you uh previously let me give you here also if you take a three mathematics for example by this time you might have already understood a11 a21 foreign if you take this diagonal this is the principle diagonal so I told you and you know that upper diagonal elements must be zero and whenever diagonal elements on the upper diagonal definitely this I value means a row number is definitely less than the column number so these elements they must be definitely zeros all these three elements of course it's a 3 by 3 so we have only three elements so all the upper diagonal elements in general should be zeros okay so now with this understanding let us try to Define what is a diagonal matrix okay so what is a diagonal matrix three diagonal Matrix okay so let us see what is a diagonal matrix so basically you all know the diagonal matrix is nothing but all the principle die means every elements other than the principle diagonal elements they should be zeros cut so how we Define that in terms of this i j notations let us check here so a matrix a is equal to a matrix a is equal to a i j for all one less than or equal to I comma J less than or equal to n is said to be is said to be a diagonal matrix is said to be a diagonal matrix is said to be a diagonal matrix if now you know one thing if the elements are above the principal diagonal they must be definitely zeros if the elements are below the principal diagonal also they must be definitely zeros so which means except the principle means of course even principal diagonal elements can be zero but you are very much sure about the upper diagonal and lower diagonal elements being zero so if the elements are lying either on upper diagonal or lower diagonal then definitely you know one thing I is not equal to J correct because in case of upper upper diagonal elements if you see I is less than J in case of lower diagonal elements I is greater than J which means in both the cases I is actually not equal to J correct so is said to be a diagonal matrix if a i j is equal to 0 for all I not equal to J so this is one very very important Matrix why I will tell because there comes one very important Concept in matrices which is called diagonalization which we'll see at the end hopefully so if you see this is the Matrix so example foreign for example so let us suppose this is the case Okay so if this is the case then you can understand one thing this is the principle diagonal here again so this is the principle diagonal so now if you check whenever the elements are not on the principal diagonal then definitely those elements should be zeros and if elements lie away from the principle diagonal means if they are not on the principal diagonal then this I and J will not be equal to any of these elements so that's why this element should be 0 this should be 0 this element should also be 0 and obviously all these elements they should be zeros clear so and even the principal diagonal elements can also be zeros but one thing that we need to make sure is all the non principle diagonal elements they must be definitely zeros clear so this is a diagonal matrix and there is one very important concept when you talk about this diagonal matrices basically okay and of course you know how to add two matrices or subtract two matrices basically so right away I'm going to talk about multiplication of matrices in the next part okay so I we have a lot of many more types of matrices but for that you need to understand certain Concepts like transpose multiplication of matrices those kind of things so that's why we are talking about multiplication of matrices for a while okay so we will go for one thing which is multiplication of matrices multiplication of matrices Matrix addition and Matrix subtraction since many of you are very familiar at this level so I'm not you know going again to that but multiplication of matrices is something which is very important so we'll see here okay so multiplication of matrices so the first property that I would like to give you is if two matrices if two matrices a m by n and B P by Q what's the meaning of this a is the Matrix of size M by N means it has Gamers and N columns B is the Matrix of size P by Q means it has POS and Q columns so if two matrices a m by n and B P by Q are to be multiplied are to be multiplied are to be multiplied then then definitely if you see this number of columns in the first Matrix should be same as number of rows in the second Matrix so this n should be definitely equal to P so why this happens because if you have a quick remembrance of how we do the matrix multiplication you can easily understand this because and how we multiply these two matrices look I will tell you let us of course you have a b c d e f and so this is a matrix basically of size 2 by 3 then definitely you should have a matrix starting with trios and any columns you can have for example say two columns if you design this then three by two so three rows and two columns so F so let us say one to 3 4 5 6 for example okay so if you see these two are equal number of columns here and number of O's here which I have stated so how you do this multiplication basically so if you see the final output Matrix is going the way of this size okay so the final Matrix if you see is going to be of this size this is of the form M by Q so 2 by 2 okay so if you see if you multiply these two matrices for example let us suppose if n is equal to p and if you multiply these two matrices the product Matrix is going to have this order of M by Q okay so that's how here it's starting with 2 these two are equal and finally it's ending with two so we have 2 by 2 here clear now let us see how we will multiply so you multiply this first element with this first element means basically to get the first element here then you should multiply the first row with the First Column similarly you will be having one more element here so to get this element since it is 2 1 a 2 1 so that's why you need to multiply this second row with the First Column that's how you get the first row of this Matrix I am repeating if you want this element for example you know this is the element in the first or second column so if you want the first or second column element in the product Matrix you need to multiply the first row with the second column okay so anyhow let us do the multiplication quickly and so if you do the multiplication a into 1 so a plus b into 3 so 3B plus C into 5 so 5c so this is the first First Column element of this Matrix now what about the first or second column so if you want first or second column A into two so basically two a plus 4B plus 6 this is going to be the first or second column okay next coming here second row First Column so 2 1 here is 2 1 so D plus 3 times of e plus 5 times of f similarly here you can write easily 2D plus 4 e plus six F okay many of you may feel like this is all basic stuff but I'm repeating understanding these matrices in this i j formats is really important so that's why I'm you know going for this okay now I'll ask you one very important question I'll give you time for five seconds you can pause the video here and you can try to answer let us suppose if I want to multiply these two matrices a is of size M by n and B is of size let's say e n by q p and M P and S Y P and N both are equal so let us suppose if these two are same then if you want to multiply these two matrices what are the total number of multiplications you need to do look for example to get this first element what I'm doing I am multiplying this a with this one so first multiplication I am doing again to get this 3v I am multiplying this B with 3 so I am doing one more multiplication C with five I am doing one more multiplication so basically how many multiplications I need to do if you have this Matrix of n and p basically here okay the answer is number of multiplications required number of multiplications number of multiplications are required required to multiply two matrices to multiply two matrices to multiply two matrices a m by n and b n by P is do you know how many the answer is going to be M into n into P so these many multiplications actually you need to do to get this elements okay and this is I can show you this result very easily okay but before showing the result I would like to give you one more important Point number of additions number of additions Editions required number of additions required to multiply to multiply two matrices to multiply two matrices a m by n and B and by P is is you can see is M times of n minus 1 into P so basically how did I get this Expressions it is very simple and first of all you may have one doubt with the second point if I am multiplying two matters how competitions are coming into the picture correct this may be common intuition but here if you see to add these three numbers three multiplied values you have but to add these three multiplied values you are doing two additions correct if you have three numbers you will do two additions right to get the total sum that's how so let us quickly see how we are getting this formulas for example let us suppose a is equal to because it has n columns so they should be n numbers in the first row similarly a two one a two two a two three and so on a to n and this goes on finally you know that the MLS in this Matrix so a M1 so you can see that this is a M1 a m 2 A.M and so on finally you have a and N this is the Matrix that's going to come here and what about this you know B Matrix B B is it has again NOS so let us suppose you write b 1 1 b 1 2 b 1 3 and so on it has P columns so we have written n by P so this is B one p similarly b 2 1 b 2 2 b 2 3 and so on b2p so this goes on and finally we have this n rows here so that's why n two entry and so on B and P so these are the two matrices now let us suppose if I ask you if you want to multiply these two matrices how many multiplications you need first of all tell me one thing if you multiply this matrices As A and B the order of this Matrix is going to be M by P correct so because if you see this is of order M by n and this is of order n by P and if you multiply these two the resultant is going to be having of order M into P now try to understand one thing so total number of elements in this Matrix will be M into P you know this correct so total number of elements in this Matrix is nothing but number of hours into number of columns now tell me one thing if you want to get the first or First Column element if you want to get the first or First Column element what you will do you will multiply this first row with this First Column correct so for multiplying this first row with this First Column how many multiplications you will do a 1 1 into B one one first multiplication you will do next a 1 2 into B two one second multiplication you will do a 1 3 into B three one so third multiplication you will do and so on if you keep on doing five by the time you reach a one n into b n one you are total doing n multiplications because the N elements in the first object which means for getting the first First Column element you are actually doing n multiplications all right so for getting one element you are doing n multiplications so to calculate this m into P elements how many multiplications you will do n into this MP which is nothing but this mnp clear now you can understand the same thing if you are doing n multiplications to get this first number then how many additions you will be doing obviously number of additions are 1 less than the number of multiplications so that's why if you are doing n multiplications to get this first or First Column element that means you will definitely do n minus 1 additions to add them up so for one element you are doing n minus one additions so for M into P elements how many multiplication how many additions you will do n minus 1 into MP so this m into n minus 1 into P otherwise you see is the number of additions required to multiply the two matrices that's a you know basically in the multiplication okay so this is one very important uh you know understanding from here now the second point that I would like to talk about matrix multiplication is matrix multiplication matrix multiplication matrix multiplication need not be commutative need not be commutative need not be commented so what is the meaning of this you know the community law in mathematics so basically means a into b a b you can say need not be equal to ba a b a need not be equal to ba equal to ba now how I can show this for example because I have told some you know a rule just now that matrix multiplication need not be commutative but how do I prove this to you look I can prove you for example let us suppose you have taken this Matrix let a is equal to let a is of size 3 by 2 and B is of size 2 by 4 for example then if you see then if you see these two matrices a into B is possible correct means if you multiply this B with this a this means this implies a b exist s or no here we will exist why because if you multiply in this order this number of columns in the first Matrix is same as number of words in the second Matrix but B into a if you try multiplying B into a then 2 by 4 and 3 by 2 Matrix correct so basically B A is something like this B is 2 by 4 Matrix and a is 3 by 2 Matrix so here clearly you can understand one thing that here the number of columns and number of us are not equal okay so that's why this kind of multiplication is not possible for the same two matrices so but ba doesn't exist this doesn't exist so forget about equating them or not equating them sometimes even even if a b exists BM may not even exist okay so that's why you can never confirmly say that matrix multiplication is commutative this uh you know this result is actually one good thing that shows that matrix multiplication cannot be commutative clear so this is one very important property again now third property matrix multiplication is associative okay matrix multiplication is associative is associative so basically what's the meaning of this if you have three matrices for example let us say then if you first of course all these three matrices are facilitating matrix multiplication means number of columns in the first Matrix should be same as number of uh you know hours and second Matrix all these things are holding good so whenever in such cases if you have three matrices for example then a of this BC will be same as a b of c means first of all when you have three matrices now if first you multiply BC and then you multiply here to the result or if you multiply a b and then if you multiply C the original domain same why this property is very very important actually Frankly Speaking this property is very important why I will tell you let us suppose you are some big matrices some 50 by 50 100 or 50 by 100 and some 100 by you know 50 again some matrices like this now many of you might be doing this C plus plus or coding in a general in Labs so what it happens is normally you write a code if you want to multiply three matrices you will write a code and the time of execution of this code depend upon the number of calculations number of automatic calculations the computer performed correct now let me give you one basic problem you can easily understand let us suppose you have this matrices a is 3 by 4 B is 4 by 2. C is 2 by uh you know 3 for example let us suppose so that the total product is going to be at a mathematics cut so if you calculate this ABC somehow then you are going to get a 3 by 3 Matrix now let us see the two ways for example so let us suppose in the first case you are multiplying this a to this BC and in the second case you are multiplying this A B to C in both the cases you will get ABC and both these values are also same the computer displays the same values but if you want to calculate number of operations performed by this computer then let us check here so this is equal to this BC BC is a matrix of size 4 by 3 but what are the total number of multiplications required to calculate this product VC number of multiplications required is how many 4 M and P correct so four to the eight eight is a 24. so to calculate this product BC you need 24 multiplications here and of course 4 into 1 into 3 12 additions also now this a is a matrix of size 3 by 4 and this BC is a matrix of size 4 by 3. now if you multiply these two matrices again how many multiplications you need 3 into 4 12 is a 36 so plus 36 more multiplications unit and total you need 60 multiplications to calculate this product a into b c correct so basically to calculate this product BC first you need 24 multiplications once you get BC to multiply this a with BC you need 36 more multiplications so total 60 multiplications come to this case so if you calculate a B for example for calculating a b how many multiplications you need 3 4 12 24. so 24 multiplications you need and this Matrix is a matrix of size 3 by 2 and C is a matrix of size 2 by 3 correct so now if you see the multiplication of this a b with C how many access multiplications you need this is a 3 by 2 and this is a two by three so three twos are six six these are eighteen so if you see this is 18 so total 42 so here in this case you need only 42 multiplications of course these are small numbers so that's why 42 and 60 of course still there is much variation but imagine when you are dealing with 100 by 50 of that size when you are in some research or some work you will definitely write the programming in such a way because you see the result is same in both the cases so in such cases what you write you will definitely write the code in such a way that the minimum number of operations will be done by the computer so that your execution time will be faster so using this simple manual analysis you will come to know which order to multiply and you will write the code accordingly for reducing your execution time clear so that's how in this case the number of multiplications required is less and its minimum basically okay so that's how this mnb is very important here and of course this law matrix multiplication is associative is also very important clear I hope I've uh you know cleared your questions and uh previously when I was taking some YouTube class people were asking me uh when you're going to give the content on all definitely I know there's very less time for gate 2023 so uh you know we have actually planned that if it's a single shot video so it would be very helpful for you so that cleansing will go from four to five hours if you spend you will definitely get one complete uh you know module some linear algebra numerical methods whatever completely okay so let's continue now I would like to give you one very important property in real numbers let us suppose if you get some expression X into Y is equal to 0 you write X is equal to 0 or Y is equal to zero okay because in numbers you know that for the product to be 0 at least one of them or maybe both can be zero but in case of matrices you can see the product of the product of two non-zero matrices the product of two non-zero matrices can be aziomatics can be a geomatics look this is very strange correct I will show you with the help of an example again you can understand this so let a is equal to some value a zero zero zero so this Matrix is not a geomatics because this a smaller what I am taking here is non-z okay so small a not equal to zero so definitely this Matrix you cannot call this a geomatics basically you cannot call this a null Matrix because this is non-zero and B is equal to zero zero zero B and here also B not equal to 0 so here this is also nonziomatic skirt so these two are non-zero matrices because they have at least one element as non-zero but if you calculate this a b let us see what it turns out a 0 0 into 0 0 0 B so now if you calculate this the outcome is going to be 2 by 2 a into 0 plus 0 into 0 this is 0 a into 0 plus 0 into B this is 0 since these two elements are zeros definitely these two elements will also be zeros so if you see this is a null Matrix okay so this is null Matrix this is null Matrix so that's how you have seen clearly these are two non-zero matrices but still the result is a zero Matrix which generally don't happens in real numbers so that's why this law or you can see this what do you say this uh you know result you can say this result is also one very important result because generally you have this intuition that for the product to be zero at least one of them should be zero but in matrices you did not have that clear now let us continue the types of matrices or before continuing I would like to give you uh one more Point okay so transpose of a matrix this is one very important thing transpose of a Matrix transpose of a matrix so what is transpose of a matrix so transpose what's the meaning of this basically reflection okay so if you interchange you need to actually interchange them so a matrix B is equal to b i j for all one less than or equal to I less than or equal to n and one less than or equal to J less than or equal to m is said to be is said to be the transpose of a matrix is said to be transpose of a matrix the transpose of a matrix capital A is equal to a i j for all one less than or equal to I less than or equal to M and 1 less than or equal to J less than or equal to n if a i j is equal to b j i b j i so this is how we Define transpose of a matrix and normally in simple terms you know if you interchange a rows and columns then definitely you will get the transpose of the given Matrix correct now why this definition is very important because if you see very soon after the single shot videos will come up with a problem solving uh you know series basically so now if you see why this definite in that in that basically in that you'll see lot of questions that transpose is given in these notations okay so now if you see what is the meaning of this let us try to understand first of all you need to understand one thing this Matrix capital B is of size n by m it's not M by n it's n by m because row numbers can go from 1 to n and column numbers can go from 1 to M similarly this Matrix a which is the initial Matrix it has M by an order so definitely you know one thing when you have a matrix of order M by n when you have a matrix of order M by n then if you interchange rows and columns then the number of force will become number of columns and the number of columns becomes number of rows so this Matrix B will be of this size n by m correct for example let us take some simple values 2 and 3 let us take then this becomes 3 by 2. for example let me take one Matrix if I take this Matrix 2 by 3 so 1 2 3 4 5 6 then you know if you want to call this B as a transpose which is denoted by this so then you know you need to Interchange horizontal column so this First Column can be interchanged as the first row then second hello and third so now if you see this is of size 3 by 2 because this is of size 2 by 3 it's quite of yes cut now what is the meaning of this Matrix look if I take b i j means if I take for example b 2 3 or B22 element B22 then definitely this is going to be same as a22 or if I take then definitely it is going to be with A1 so that's why I told you that a i j is equal to b j i means if you are taking second row second or you can say first or third column element in B you should take third or First Column element okay so these two will be equal that's what I have defined in slider complex terms because this is generally important for get examination okay and you can write one more statement it is denoted as it is denoted as it is denoted as B is equal to a transpose t on the superscript of T of a basically here okay so that's why I hope you understood this I'm repeating whenever you are making a transpose the order of the Matrix gets interchanged it means basically if this is of size M by n then this becomes of size n by m as you have seen and obviously if you find out element a i j this element will be same as b j i with the order and row number reversed because you have interchanged the matrices that's the reason and this is how you actually Define the transpose of a matrix and this is very uh you know interesting thing because transpose means basically have also become columns and columns become others normally if you see the meaning of transpose this if you have any Matrix and if you do transpose of it the points will actually come as a reflection about the line y equal to X basically okay so for example if you have one two Matrix then if you transpose it it becomes 2 1 and that point one comma 2 and 2 comma 1 if you take these two points they are nothing but the reflection of each other about the line Y is equal to X generally okay so that's why uh you know this transpose is one very important uh you know uh operation in matrices now if you uh C properties of transpose I would like to give some properties of transpose properties of transpose properties of transpose okay so now the first property let us suppose if you transpose any Matrix even number of times for example then you will get the same Matrix correct let us suppose if you transpose once okay then you will get the transpose of that Matrix now let us suppose if you transpose it again means if you apply the transpose two times then what happens first time when you do the transpose you will get the interchange of the rows and columns but second time when you enter changing again you will again get the original Matrix okay so even number of Transformations give you the same Matrix odd number of Transformations give you the you know the transpose Matrix okay so second if K is any scalar if K is any scalar so scalar means some number okay some constant if K is any scalar then k a whole transpose is actually equal to K times of a transpose so this result is again very important because you can never transpose any scalar okay so for example if you see 2A for example if I take this Matrix a is equal to 1 2 3 4 simply I am just denoting it so this implies if I calculate 3 times of a I need to multiply 3 to all the elements so if I do the three a whole transpose correct so this Matrix is nothing but 3 times of 1 3 2 4 and this is 3 into this is a transpose but scalars this callus they never undergo any transformation basically okay so that's what we have here then third you can see some a plus or minus B whole transpose if we have this then this is equal to a transpose plus all minus B transpose means basically if you take plus here here also it is plus if you take minus here this is also minus you can check all this with simple examples and one very important property that I am going to give you now is a b whole transpose or you can write C also whole transpose is equal to C transpose B transpose into a transpose so all these are basically some nice results of additional multiplication of transposes of matrices okay and this result is very important fourth result especially we'll use after some time in this lecture so now if you see this fourth one is a pretty much good property of matrices actually and then you have these are the four important properties that generally sometimes they ask you in examinations and also in uh you know General University exams now coming to the continuing to the types of matrices we have defined three types of matrices what are the lower triangular upper triangular and also diagonal matrix correct so let us continue so again fourth if you see we'll Define something called scalar Matrix scalarmatics so in simple terms if I want to explain you what is a scalar matrix it's basically a diagonal matrix but this time all the diagonal elements are equal like for example you see the case of identity Matrix all the diagonal elements are equal okay so that's how this scalar Matrix is basically done so a matrix a is equal to a matrix a is equal to a i j for all 1 less than or equal to I comma J less than or equal to n is said to be scalar Matrix is said to be scalar Matrix if a i j is equal to some constant k for all I is equal to Z and 0 for all I not equal to J so you can see this time this a i j is given as some function of I comma J but whenever I is equal to J means if element is lying on the principle diagonal then it should have some fixed value but when elements are non-diagonal elements then definitely they are zeros and you can see this is something like this example if I take 3 by 3 then you can see zero and zero zero K so this is how you can get a scalar Matrix basically all the principal diagonal elements are same you can write here a diagonal matrix a diagonal matrix a diagonal matrix with all the principle diagonal elements with all the principle diagonal elements with all the principal diagonal elements being same with all the principal diagonal elements being same actually here okay so now if you see if you can understand one thing carefully if K is equal to 1 then this Matrix is called identity Matrix okay because you know in identity Matrix you have the principle diagonal elements one and all other elements zeros so when K is equal to 1 because this can be any constant so when K is equal to 1 this Matrix is called identity Matrix and if K is equal to zero foreign if K is equal to 0 this is called null Matrix null Matrix okay so this K is equal to 0 this gives you null Matrix actually here okay so these are two special cases of this scalar Matrix so uh basically you deal with identity Matrix everywhere when you talk about inverse of the Matrix orthogonal all these cases and null Matrix is again the common uh you know uh what do you say additive means basically if you see uh additive identity uh Matrix is basically this K means if you add aromatics to this Matrix null Matrix you again get the same Matrix okay so this is calimatics now I'll go for one very important Matrix which is called symmetric Matrix symmetric Matrix you can tell by the name itself The Matrix is symmetric about its principle diagonal so now you see what is a symmetric Matrix okay so in simple terms you know a is equal to a transpose let us write it in terms of idea notations a matrix a matrix a is equal to a i j where one less than or equal to I comma J less than or equal to n I comma J less than or equal to 1. is said to be symmetric is said to be symmetric if a i j is equal to a j i means what does this mean actually here if you see this a i j is equal to a j i means if you take any element symmetric about the uh you know basically symmetic about the principle diagonal then the elements values are same okay so example I would like to give you example a is equal to for example you can write 2 3 4 then this is also three four five six then this is also 6 7 let us suppose if you take this then what happens here if you see carefully this is the principal diagonal so this is the principle diagonal so if you see this element for example if you say this element this element is on the second row I mean sorry the first or second column so that element three which I have ordered off is a one two so now this a12 will be same as for example if I write this a 1 2 now if you enter change this ji values this should be equal to a21 correct so if you see this is also same here so these two elements will be same generally similarly these two elements similarly these two elements so about the principle diagonal if you take the corresponding reflection then the reflection value and the actual value both will be same actually okay so this is how you define a symmetric Matrix and in general you can write basically this a is equal to a transpose clear so that's how this is one very important uh you know thing in matrices now I would like to give you one again one very important type of Matrix skew symmetic Matrix okay this Matrix is very very much important okay why I'll talk about it when I come to the eigen values okay so see here six skew symmetric Matrix geosymatic Matrix so again in simple terms you know a is equal to minus a transpose but let us write it in either notations a matrix a matrix a is equal to a i j for all 1 less than or equal to I comma J less than or equal to n foreign Ty Matrix geometric Matrix if a i j is equal to minus a j i so basically if you see again here the reflection value should be same but of course with a opposite sign okay so numerically these both are same values in terms of magnitudes but in terms of you know signs they are actually opposite example I can tell you one thing example you can get many examples example a is equal to zero zero zero if you hide 2 here then this is -2 if you write minus 3 then this is plus 3 if this is 4 then this is minus 4 and you can say like this so here if you see this is one example of a schematic Matrix so for example across the principle diagonal this is the principal diagonal so from this principle diagonal if you take element a one two then this a 1 2 and a two one means these two elements if you see magnitude wise they are same but there is a sign difference between them now you can understand one thing from this example itself all the principle diagonal elements are 0 so why this is happening look I will tell you I'll give you some two nice points here first point principal diagonal elements or you can write all the principal diagonal elements all the principle diagonal elements of SQ semantic Matrix of SQ symmetic Matrix r0 all the diagonal elements of SQ symmetric Matrix are zero now why this is happening because you know in general if first Q symmetic Matrix you know for SQ symmetric Matrix for SQ symmetric Matrix or sq symmetic Matrix as I have told you a i j is equal to minus a j i now let us suppose if the diagonal this element falls on the principle diagonal okay so this element falls on the principle diagonal means for I is equal to J for I is equal to J this a i i is equal to minus a i i so this implies 2 times of a i i is equal to 0 which means your a I I should be equal to 0 means this element value should be equal to 0 whenever it falls on the principle diagonal because obviously 2 is not 0 okay so here if you see whenever element falls on the principal diagonal the magnitude twice they should be same but science it could it should be opposite okay but these two they denote the same you know values basically if I am talking about 1 1 this is the only element correct so if you see when I is equal to J this principle diagonal element should be equal to the same principle diagonal element so from this simple analysis you know all this principal diagonal elements of SQ symmetric Matrix is 0 and second point I would like to give you here second point I would like to give you here the sum of all the elements the sum of all the elements the sum of all the elements of SQ symmetric Matrix the sum of all the elements of SQ semantic Matrix of SQ symmetric Matrix is zero basically this is zero clear if you see because for every element that you find there is definitely a element with a opposite sign correct so when you add these two you'll get zero similarly when you add these two you'll get 0 when you add these two you'll get zero for every element that you find you will definitely find a reflection with a opposite sign so when you add these two elements you will get 0. so similarly every total sum of this uh elements in this Matrix turns out to be 0 here clear now I would like to give you one very important you know a result coming up to this symmetric and skill symmetic Matrix look what I'll tell you every Square Matrix every Square Matrix can be decomposed can be decomposed when splitting up can be decomposed into sum of asymmetric sum of is symmetric symmetic oh I'm sorry I think it's base limitation is sum of a symmetric and SQ symmetric Matrix asymmetric and a skew semantic Matrix now you see how it happens look I'll tell you so if you take any basically what I am telling you if you take any Square Matrix then this Square Matrix can be written a sum of asymmetric Matrix and SQ symmetric Matrix look how it happens can I write if I take a square Matrix a this can wait as half times of a plus a transpose and half times of a minus a transpose can I write this because I can write this because if you simplify the things this plus a transpose by 2 and ah this minus a transpose B2 and plus a transpose by two they cancel each other so a by 2 plus a by 2 will again give you a so now if you see this if you see this Matrix if you see this Matrix this Matrix is always symmetric symmetric and this Matrix is always skew semantic okay excuse him attack now how come you know this because look tell me one thing let us suppose if a matrix a is symmetric for example okay okay let us take some Matrix B symmetric if B is symmetric then any multiplication of B is also symmetic means basically if I take 3 times of B 5 times of B 10 times of B whatever I take 10 B 5 b 3 B whatever you take all these are also symmetric matrices similarly if B is Q symmetric then again this is ab5 B7 way all these things are Auto B 4B anything all these things are schismatic so how to identify whether a given Matrix is symmetric or excuse symmetric Matrix look I'll tell you let C is equal to let C is equal to a plus a transpose okay so what does this imply if you take C transpose this gives you a plus a transpose whole transpose so this is equal to in properties of transpose I have given you if you have sum of two matrices whole transpose you can write the sum of individual transposes so this is equal to a transpose plus a transpose whole transpose so this is a transpose plus this is a and this is again same as C because C is same as a plus a transpose or a transpose plus a both are same so C transpose is finally ending up with C so that's why this Matrix C is actually symmetric okay so this implies C transpose is equal to C so this C Matrix what you assumed a plus a transpose is always symmetric similarly if you take this D if you take this D let D is equal to a minus a transpose so this implies your D transpose is equal to a minus a transpose whole transpose so a transpose minus this a transpose whole transpose if you write then this is going to be a transpose minus a and if you take a minus sign this is coming out as a minus a transpose which is nothing but minus D so your D transpose is equal to minus D so which means this implies that this Matrix is skill symmetic skew symmetry okay so this is symmetric Matrix so this is basically symmetric and this a minus a transpose is always Q symmetic okay so if you take any Square Matrix you can definitely split this into sum of two matrices out of that one is symmetric and the other being excusometer okay so I hope we are clear with this uh result that every Square mat X can be split into sum of asymmetric and a skill symmetric Matrix okay now let us go for one way next important result that normally these statements when they ask you in examinations people get very Affair but they are simple look if a comma B are symmetric matrices of same order symmetric matrices of same order symmetric matrices of same order then I'll give you the result first then I'll show you how it's happening first is a B plus b a is always symmetric is always climatic a B minus b a is always q symmetic but provided these are symmetric matrices individually okay so is always skew symmetric skew symmetric now how come this is possible look I will tell you you did not actually remember these results but you can actually show them so firstly they are given a and b are symmetric material so given a transpose is equal to a and B transpose is equal to B now let us consider the first case a B plus ba okay so when you consider the first case let us see what's happening here so the first case is let C is equal to a B plus ba so whenever you want to check whether Matrix is symmetric cos Q symmetic just simply consider the transpose of it C transpose is equal to a B plus b a whole transpose so this is nothing but a b transpose plus b a transpose now I have told you the multiplication of matrices basically so if you apply transpose of any multiplication then the result is going to be like this B transpose into a transpose plus a transpose into B transpose so this is equal to since these two are individually symmetric matrices I can replace this B transpose with b and a transpose with a similarly here so this is going out to be BA Plus eav okay so this result I can write as a B plus b a which is again equal to C so you have started by considering C transpose and finally you ended up that c transpose is same as C so this Matrix a B plus b a is always symmetric here one caution I have written BA Plus a b s a b plus b a not because that b a should be equal to a b no that's not the case but since the summation is commutative in any way so this is B A plus a b or a B plus b a both give you same answer but not this B A should be equal to this a b this need not happen Okay so please be sure with this now second let C is equal to or D you can say a B minus b a then what this tells you D transpose if you take a b minus b a whole transpose so this is a b transpose minus B A transpose which is equal to B transpose a transpose minus a transpose into B transpose because you have seen a b whole transpose is ah you know the transverse individual transpose multiplication but taken in a have us order so this is equal to ba minus a b so if you take a minus sign here you will get a B minus B A which is nothing but minus D actually so this ends with d transpose is equal to minus D so definitely D is excuse symmetric Matrix and what is D we have taken this A B minus B A okay but this transposes we are able to replace in this form because it's individually given that both these are symmetric Matrix is clear if this individual symmetricity is not given then definitely you need to you cannot substitute this B transpose as B A transpose as a sometimes they'll give Q symmetric here then again the process needs to be same but in place of B transpose let us suppose if they give a and b as Q symmetic matrices then here you will put B transpose equal to minus b and a transpose is equal to minus a okay so that's how we actually you know uh generate the results and these two results are very much easy now you need not remember them actually they are very easy to you know to show them it's just some to the basic properties if you apply you will get the answers clear I hope I'm clear till this point so this is a single short video so by the completion of this video you will be able to get complete knowledge on this linear algebra which is very you know uh very good for gate examination okay so and of course this video I'm inclined towards uh gate syllabus so that's why I'm dealing with this okay now once you are with this now I would like to give you some more properties okay so now some more types of matrices let us give so how many types I have given you till now so all these are properties six types I have given you so let us go for the seventh type so seventh orthogonal Matrix orthogonal Matrix so what is an orthogonal Matrix so let us see a matrix a matrix capital a is said to be orthogonal is said to be orthogonal is said to be orthogonal if a times a transpose is equal to a transpose times a which is equal to Identity Matrix okay of same size of course so a into a transpose this a into a transpose is equal to a transpose into a is equal to I now let us see here uh okay I hope mic is good yeah mic is good so no issue so a into a transpose is equal to a transpose into a is equal to I so let us take one example and let us see example a is equal to for example say cos Theta minus sine Theta 0 sine Theta cos Theta 0 0 1 so let us suppose this is some Matrix now if you multiply this Matrix with its transpose so a into a transpose is equal to a into a transpose is equal to cos Theta minus sine Theta 0 sine Theta cos Theta 0 0 1 now multiply the transpose of this Matrix so if you do the transpose it's like cos Theta sine Theta 0 minus sine Theta cos Theta 0 0 1. now if you multiply these two matrices you can easily understand that first of First Column element if you multiply this first with this First Column cos Theta into cos Theta cos Square Theta everything Plus minus sine Theta into minus sine Theta sine Square Theta plus this 0 into 0 0 so the first First Column element is going to be cos Square theta plus sine Square Theta that's it okay now first or second column element cos Theta into sin theta plus minus sine Theta into cos Theta so this is cos Theta sine Theta minus sine Theta cos Theta which is 0. next third so the first row with the third column cos Theta into 0 minus sine Theta into 0 0 into 1 so total 0 so this is how you have this next second or first column sin Theta into cos theta plus cos Theta into minus sin Theta so this is sin Theta cos Theta minus cos Theta sine Theta and that gives you 0 directly now second or second column sine Theta into sine Theta sine Square theta plus cos Theta into cos Theta cos Square Theta so again you got here sine Square theta plus cos Square Theta okay next second or third column again it is 0 directly now third or First Column these two I have zeros one into zero zero this is also zero and finally here you have one into one this is one so sine Square theta plus cos Square Theta is always one irrespective of the Theta value so 0 1 0 and 0 0 1 which is actually an identity Matrix now if you want more examples you can do one thing whatever you can put you can generate a matrix by putting different values of theta let us suppose if I put Theta equal to 0 let Theta is equal to 0 then I'll get 1 Matrix that b is equal to cos 0 1 sine zero zero zero this is 0 then this is 1 this is zero zero zero one so I'll get identity Matrix and identity Matrix is again one uh you know one of the orthogonal Matrix if I put Theta is equal to some 90 degrees for example then you'll get 0 minus one zero one zero zero zero zero one that's again one more orthogonal Matrix so you can if you want to generate n number for orthogonal matrices you can just go for some n uh values of theta whatever the value of theta you put for this particular configuration you always end up with an orthogonal Matrix clear so orthogonal Matrix is that Matrix which when you multiply with its transpose you always get an identity Matrix okay and this example I hope it's uh clear to you let me have some water foreign okay so let's continue so this is orthogonal Matrix now I would like to give you one more important Matrix which is involutely Matrix Matrix Matrix a matrix capital a is said to be invaluated involutely if a square is equal to I which means if you square any Matrix you will get the identity Matrix of same order and for getting this this a need not be an identity Matrix there can be some different matrices okay where if you do a square you will get I value let us see how we get this you know many times I'll be looking downwards because I have the system here so if there is some issue with the mic I will get to know okay otherwise it's recording you know there is it's not a live class so I don't know what's happening so anyhow look here a matrix capital A is said to be invaluated if s square is equal to I let us see how we get this I'll give you one simple example a is equal to uh four minus 1 let us skip so if you keep 4 minus 1 then this is uh 15 and this is uh you know two times of this so minus four so let us suppose if you take this example it is a simple example for example okay now if you calculate the square if you calculate the square 4 minus 1 15 minus 4 and this you know 4 minus 1 15 minus 4 so if you calculate this product you can see four fours are sixteen minus 1 into 5 15 is 16 minus 15 which is 1 so 4 into minus 1 is minus 4 plus minus 1 into minus 4 is plus four so minus 4 plus 4 this is zero fifteen fours are sixty plus minus 4 into 15 is minus sixty so sixty minus sixty again this is 0 15 into minus 1 is minus 15 plus minus 4 into minus 4 is plus sixteen so sixteen minus 15 this is again one so if you see this Matrix if you see this Matrix if you see this Matrix is an ah you know basically invaluative Matrix so I would like to give you one very good property here sometimes they give you that let a is equal let a is an invaluative Matrix and they give you some complicated elements inside but and they'll ask you for a inverse okay so we'll talk about how to do an a inverse simply and all but right now if you see in this Matrix there is one very special property that a inverse is basically again sorry a inverse a inverse is basically equal to a okay so what happens in that case is whenever you uh you know whenever you with all the difficulty when you try to calculate a inverse finally you try to get the same you know Matrix as the inverse Matrix okay and even after getting the same Matrix many of you will have this doubt that uh have I done some on calculation or what I have got the same question Matrix but this is actually the case with this Matrix the inverse of this Matrix will be same as the Matrix itself okay that's why if you if you observe you can tell clearly you know a square is equal to I is what you got correct so a into a is equal to I but generally you know this I can also be written as a into a inverse okay so that's why if you comparison only these two matrices they will be same of course okay so that's why this a inverse is equal to a in this case clear so let us go for the next things so this is involutely Matrix now I will give you one again some good Matrix which is idem potent Matrix item potent Matrix okay so what is this idea potent Matrix look I'll give you a matrix capital a is said to be idem potent Matrix is said to be item potent Matrix add important Matrix if a square is equal to a if a square is equal to a here okay so look if s square is equal to a now try to understand uh so basically if you try to square any Matrix you will get the same Matrix let us see we can write to any example here example a is equal to again let us take 4 minus 1 here for example so if you take 4 minus 1 this any multiplication for example 3 times you write so 12 minus 3 for example so if you take this Matrix okay all this matters why I'm talking because they have certain properties in determinants okay so in determinants if I say the determinant of an orthogonal Matrix is plus or minus 1 you should know what is orthogonal matrix by that time okay similarly if I talk something about determinant of an item product Matrix you should know what is at important Matrix so that's why these matrices I am discussing now and of course there are many gate questions on this types of matrices okay which will come to know later in the pyq's solving sessions so anyhow let us come here so 4 minus 1 12 minus 3 so this implies if you calculate a square for example 4 minus 1 12 minus 3 and 4 minus 1 12 minus 3 so if you multiply this a square is going to be equal to 4 4 16 plus minus 1 into 12 minus 12 so 16 minus 12 is 4 again so minus 4 plus 3 it's minus 1 48 minus 36 12 minus 12 plus 9 minus 3 so this is again equal to a okay so this is how you get this Matrix which is this item potent Matrix actually here okay because I will talk about some properties of this Matrix again when I come to the uh you know determinants actually okay so now let us uh go ahead and what I would like to discuss one last type of Matrix before I go further determinants actually okay you have seen transpose of the Matrix how we need to identify whether the given matrices are symmetric matrices Q symmetric matrices we have seen all these things okay now we'll go for one important topic which is determinant but before going for the determinant we have one last type of Matrix that of course we have many types but uh in Gate point of view I would like to discuss one particular Matrix at this point which is actually a oh which is actually your uh nil potent Matrix you can say nil potent matics so what is this nil potent Matrix let us see here in non-ziomatics in non-zero Matrix capital a foreign Matrix if there exist a value n a value n such that such that n belongs to Z Plus means basically what's the meaning of this N is a positive integer okay so N is a positive integer and a power n is equal to a null Matrix basically here okay so means whenever you take any non-zero Matrix but some value or some higher power of that Matrix is an important Matrix then you call that Matrix as a nil potent Matrix okay now let me see what is the case example a is equal to so let us take 2 minus 1 and 4 minus 2 for example let us say okay fine good enough so 2 minus 1 and 4 minus 2 for example if you take this uh just in check so two times of this okay fine yeah 2 minus 1 and 4 minus 2. now if you calculate this a square you see this is a non-zero Matrix but if you multiply this Matrix to itself 2 minus 1 I'm dealing in 2 by 2 so that calculations will be easy the same can hold good for 3 by 3 is also 4 minus 2 so a square is equal to now if you multiply two twos are 4 plus minus 1 into 4 is minus 4 so 4 minus 4 0 2 into minus 1 minus 2 plus minus 1 into minus 2 is plus two so minus two plus two zero four twos are eight plus minus two into four is minus 8 0 4 into 1 minus 1 minus 4 plus minus two into minus 2 plus 4 which is again 0 here so if you see this a square is a null Matrix but actually a is not an all Matrix okay this is again in evidence to that result what I have given there that product of two non-zero matrices can we still have zero Matrix okay so that's why you can see this is the case and here I would like to add one more point because uh you know it's a good point the least value of n the least value of n for which for which this a power n is equal to this null mat X is called is called index of the Neil important Matrix index of the nil potent Matrix okay so this is called index of the nil potent Matrix now try to understand one thing because if a square is 0 then definitely from next onwards every Power will go to zero a cube a power for a power five all these things will be zeros because a cube can wait as a square into a so a square is 0 a cube is also zero so all this goes on but 2 is the first time this whenever this n is equal to 2 this is the first time when a power n becomes 0 so index of this Matrix is actually two clear for some matrices you can have the four what can be the depending on the type of Matrix but here right the right now this is called this uh you know nil potent Matrix basically and this value the least value of n for which this a power n turns out to be a null Matrix for the first time is called index of this nil important Matrix okay so with this we finish this first part of this video like uh types of matrices and basic Matrix algebra multiplications editions or you can see some types of matters a symmetric matrices or nil potent orthogonal all these things okay so now next we'll move on to the second part of this video which is uh the talk about determinants okay so let us go ahead and let us discuss the concept of determinants determinants determinants determinants or determinant you can say now if I ask you for example if I ask you if this is a matrix and if I ask you what is the determinant of this Matrix okay so immediately you start telling a D minus BC that a into D minus B into C But If You observe the question carefully I didn't ask you what is the value of the determinant I asked you what is determinant correct so you should definitely know because even this question I'll tell you one of the experiences that I had recently have you know a bit of a BRC interview also so now I'll tell you one thing in every interview is General in technical interviews there will be one faculty from one you know interviewer from maths actually for mathematical background and they'll try to ask you some questions so you prepare lot of things in mathematics and you go for the interview but finally they ask you what is determinant and you'll end up with telling the value of the determinant okay so that's why this understanding of the concept of determinant is actually very important okay so now let us see what is this determinant actually look I'll define the definition formally and then we'll see what's happening here okay and you know whenever you are trying to calculate it determinant you let first term minus second term plus third term means between every terms your your actually alternating the plus minus sign why you are alternating this plus minus sign okay we'll see all these things in this uh you know part look so determinant the sum of the sum of product of elements product of elements of Arrow or column of a matrix of a matrix with the with the corresponding with their corresponding cofactors with their corresponding cofactors so what are these cofactors and what is this definition basically product of some of there are different things correct so let us see what we are actually talking about okay for and to make you understand this I'll go with the help of it Matrix so that you will understand okay so look here let us suppose I have taken this Matrix let a is equal to a 1 1 a 1 2 a 1 3. a two one a two two a two three a31 a32 and hey actually if I have explained it with the help of two by two we have actually got what is the definition of a minor or something but anyhow let me show you with this now if I see this Matrix then there are Total Line elements so now we write something called minor of a11 for example okay so if I ask you about this minor of a one one then see this is the element containing a11 so this is the column which is containing a11 and this is the O which is containing A1 so if you delete this row and column from this Matrix you will get some sub Matrix submatics is basically some some small part of the Matrix okay so from this Matrix if you delete this column and this so then what happens here you will end up with this you know value that a22 and a so if you see this value is nothing but a two to eight minus you need to you know uh understand one thing here I am directly going with the guess of 3 by 3 that's why I am not explaining you why you need to multiply these two and then with the minus sign these two okay if I have started with the example of 2 by 2 you might have easily understood but again I know that uh at this level you'll be knowing how to calculate the determinant of this I am just waiting directly okay but actually I should start the discussion with the two by two matrices then you will come to know why I am multiplying this cross minus this okay but right now I am assuming you know how to do this so I am doing it evidently now similarly if I want to write minor of some other element for example minor of uh you know a one two let us suppose I am adding minor of a one two so if I want to write minor of a 1 2 then this is the column and again this is the row that is containing a12 so this is left with foreign if I delete this column and this so I have a21 a23 1 and a double D so if you simplify this a21 a33 minus a23 a31 okay next why I am doing I want to wait for three elements minor of a one three also I'll write okay so minor of a13 also I'll write so now you see here minor of a13 if I take this uh you know elements this is the column containing and this is the row containing a13 so if you delete this column and the first row you will end up with thing like this so uh determinant of a11 oh sorry a21 a22 1 a32 so this is equal to if you do this again a two one a32 minus a22 times a31 okay so this is how you have this now these are the three elements which I have written intentionally but if you want you can write for all the nine elements so if you write for all the nine elements you will get nine different minus okay so now let us see what I Define by a cofactor so cofactor of cofactor of an element an element a i j this is equal to minus 1 whole power I plus J times minus of a i j minus of a i j now if you see the cofactor of a 1 1 a 1 to all these things you will get a different elements with a plus sign minus sign change now try to understand the definition now if you see I can get look this is the first row for example I have calculated the minus of the first a 1 1 A1 to a one three so now try to understand one thing if I calculate the minus then definitely I can calculate the cofactors of these elements also correct so what will be the cofactor of this this minus a 1 1 into minus 1 whole power one plus one similarly cofactor of a12 is equal to a one two I mean this minus of a one two into minus 1 whole power one plus two into minus one whole power one plus three similarly here now try to understand one thing if I want to calculate the determinant that ta can be written as now look at this definition the sum of forget the sum of first of all product of elements of Arrow or column with the corresponding cofactors look let us suppose if I take the first row then what I should do is I should multiply this a 1 1 with the cofactor of this a one one then I need to sum this to the product of this a one two with the cofactor of a 1 2 plus into cofact of a13 means basically if you see this data is equal to a 1 1 into cofactor of a11 plus a 1 2 into cofactor of a12 Plus into cofactor of a13 look here I have taken the first row but it's not magnetic that you need to take always the first row you can take second or third or anything or you can also take First Column second column third column anything but you need to multiply the same cofactor correspond to that element this must be done carefully okay so now let us check here this is equal to a one one into now cofactor of this a 1 1 will be nothing but minus 1 whole power 1 plus 1 into minor of this a one one correct so minus 1 whole power 1 plus 1 into minus of a 1 1 let us suppose if I write this as m one one this is m one two this is m one thing minus basically okay so now you try to understand one thing Plus a one to into minus 1 whole power 1 plus 2 into m 1 2 plus a 1 3 into minus 1 whole power 1 plus 3 into m one three so now let us substitute the values if you substitute the values what you will get a 1 1 times of this minus 1 whole power 1 plus 1 is nothing but plus one so first of all I want you make you understand that why you are putting a plus minus n alternatively because this gives you plus sign this gives you minus sign again this gives you plus sign if you have four by four for example then 1 plus 4 will give you a minus sign that's the reason why plus and minus signs keep on alternating depending on the sum of minus 1 whole power this one plus some sum basically here okay so if you see depending on this summation here you have a plus sign and here you will get a minus sign again here you will get a plus sign here you will get some minus sign that's the reason of adding alternative plus and minus in case of determinants okay so this if you write this foreign minus minus uh then again here you have a plus so plus a 1 3 times off there you can write 8 1 a32 minus A2 to A1 so basically this is how you actually write the determinant value when they ask you for the determinant of the given Matrix okay so when you have this is the determinant value now I would like to tell you one thing if you multiply the cofactors of the same elements with the corresponding cofactors you will get the determinant value means if you multiply this o elements with the corresponding cofactors you will get determinant now let us suppose if you multiply the elements of a row with other cofactors means basically you are taking elements of one row and you are multiplying these o elements with cofactors of some other row then what will what you will get the result the answer is going to be 0 okay look I will show you how it's coming basically how it's coming I'll show you with the help of a two by two look let us suppose you have taken some two by two Matrix which is a b c d now if you write the minus in half I would like to declare the inverse of this if you want to add the inverse you should always write this one by determinant a D minus BC and now interchange these two terms so d and a comes here and of course these two has to be replaced with the minus sign minus B minus C you will see shortly how this is coming and all so a inverse the simplest form to a inverse is this one by a D minus b c o a directly then you swap these two means basically you interchange these two diagonal terms so d and a comes here then U plus these two with a minus sign okay so anyhow if you see look I will show you if you take the first o means m one one minor of first or First Column element if you take this element then if you delete this o and this column then D is the thing which is left over so d m one two so if you take this element B now if you delete this o and this column you are left with this C so this is C next M to 1. so if you take this C for example if you delete these two you have a b and M 2 2 will be a correct so what are the cofactors so if you want to talk about cofactors then cofactor of 1 1 you can write capital A 1 1 cofactor of 1 1 is equal to 1 plus 1 into D which is equal to D ofactor of 1 2. into this C which is minus C ofactor of 1 3 or OK it is a two by two so two one is equal to minus one whole power 2 plus 1 into B which is minus B a22 is equal to minus 1 whole power 2 plus 2 into a which is equal to a okay so these are the cofactors now this is the cofactor of 1 1 this is the cofactor of 1 2 this is cofact of second or first column this is cofactor of second or second column now as I told you you all know that that a if I ask you this you will directly tell me a D minus BC but how this a D minus BC is coming if you multiply first row with the corresponding cofactors you will get correct so Delta can be written as first two elements multiplied with the first row cofactors so first row element a what is the cofactor of a you can see cofactor first of First Column element is d then plus second element B into what is the cofactor of this first or second column element minus C so you can see this is nothing but a D minus BC which is the determinant of this Matrix next you can calculate that a using another method also use by taking some other also for example let us suppose if I take using this column BD any column you can take b d let us take so first B B into then what is the cofactor corresponding to this B cofactor transponding to this B this is first or second column element so first or second column is minus C Plus this element is D and what is the cofactor corresponding to D last element is a so you can see this is nothing but minus BC plus a d which is nothing but a D minus BC again you will get the determinant so my intention is basically to tell you that if you multiply elements of arrow with their corresponding cofactors or maybe column with their corresponding cofactors you will get determinant value okay now what happens if you multiply the cofactors of some other with the actual some other elements okay for example let us try to multiply this so with these cofactors if you try to multiply let's see what happens a into so cofactor of C is how much minus B oh plus b into what is the cofactor of d perfect of D is a so if you see a b and minus a b the answer is called to be 0 okay so if you multiply elements uh means if you multiply element software with other cofactors you end up with having a 0 this is a two by two case it's very simple so I have shown you using this two by two case but actually you can show for three by three also so basically my intention is to tell you that if you multiply elements it means basically if you take the sum of this product of elements with their corresponding cofactors you will get the determinant and if you multiply some other elements cofactors you will get this 0 now I want to tell you one thing here look look let a one one a one two yeah one thing a two one a two two a two three one a32 our cofactors of a our cofactors are cofactors of a then what can I eat I can form some cofactor Matrix correct so cofactor Matrix cofactor Matrix cofactor Matrix is going to be something like this perfect of the first or First Column element first or second column element first or third column a21 a22 a23 a31 a32 and a33 okay so let us suppose this is the cofactor Matrix means basically you have calculated cofactors of all the nine elements now let us suppose if you take the transpose of this why I am taking transpose you will come to know when I multiply okay so let us suppose I am taking transpose so transpose foreign Factor Matrix transpose of this cofactor Matrix is called is called adjoint Matrix transpose of this cofactor Matrix is called a joint Matrix now let us see means basically if you want a joint Matrix a joint Matrix this is equal to cofactor Matrix cofactor Matrix full transpose okay now let us see what is this adjacent adjoint Matrix basically so this implies a joint Matrix a giant Matrix or generally you write this as a d j of a joint of a so a joint Matrix of a is nothing but this transpose of the cofactor Matrix which means a11 a12 a13 similarly a two one a two two a two three one yeah three two okay so let us suppose this is the cofactor I mean this adjoint Matrix now let us see what happens I want to show you one very important result at this moment because that result is the main uh basic for or the concept of you know uh a joint and inverse of the Matrix action leg okay so let us see here so considering considering a into joint of a for example if you consider this a into a joint of a look I am considering this a 1 1 a 1 2 a 1 3 these are normal elements so Matrix a elements a31 a32 and a33 now let us suppose if you multiply this with this adjoint Matrix which is a 1 1 a21 or a12 and a one thing because we have transposed it a two one a two two a two three one for example so let us suppose if you have this okay so once you have this then see what's happening here if you multiply these two a into a joint of a into a joint of a is equal to look try to observe carefully now for getting the first or First Column element you need to multiply the first row with the First Column and all this means if i t a 2 2 that means this Capital A2 is nothing but cofact of this element small A2 if I had Capital a32 this capital a32 is nothing but Co fat of the element small a32 correct now let us see if we multiply if you multiply this first row with the First Column a 1 1 into Capital One means you are multiplying the element with its corresponding cofactor plus a 1 2 into capital A 1 2 plus a 1 3 into Capital a13 so if you see the first or First Column element is nothing but the sum of product of elements of this o with its corresponding cofactor so which means you are going to get determinant of a here correct so the first element that you are going to get is actually the data next if you see first o with the second row of factors obviously zero first of with the third cofactors this is zero next second dough with the first or cofactors zero second row with second or cofactors then again this is dirtier second row with third Auto factors zero similarly this is zero zero debt a so this is one very important thing and from this Matrix if you take this data constant as multiplication you will have DTA times of one zero zero zero one zero zero zero one so which is nothing but Delta into identity Matrix so if you take any Square Matrix this result is very much important that for any Square Matrix for any Square Matrix a into a joint of a into a joint of a is equal to that year times of that Matrix okay so this is one very very important equation and this equation is coming here from the definition of determinant because if you see here you are supposed to write that a only if you know the definition of data correct because you are actually summing up the product of this elements with its corresponding cofactors cut so for writing this Matrix here you should know two conditions that if you multiply elements of a or column with its corresponding cofactors you will get determinant and if you get if you multiply elements of Arrow or column with some other cofactors you will get 0 so that's how these two rules will help you out inviting this Matrix capital I mean this Modi uh this determinant of a 0 0 0 and 0 0 and you are able to form this equation with the knowledge of definition of determinant okay and why this equation is very very important because this is the equation that gives you a fundamental for the concept of inverse of a matrix okay multiplicative inverse if you say look if a inverse is multiplicative multiplicative inverse of a matrix then definitely you know that a into a inverse gives you identity Matrix correct now let us see how to calculate the inverse of this Matrix look I'll tell you for this equation we have look we have a into a joint of a is equal to Delta into I so if you multiply this a inverse on both sides a inverse into a into a joint a is equal to that is going to have constant into a inverse into I look in matrices it is very very important if you are multiplying a inverse before this here also you should multiply before this okay this is one because in numbers we do not follow this generally but in matrices you have seen the order of multiplication is of very much importance so that's why if you see if you are multiplying this a inverse here then definitely you should multiply a inverse before this I okay so now we have seen matrix multiplication is you know Associated basically cut so I can take multiplication of these two first and then these two now if you see a inverse into a is nothing but I into a joint a is equal to Delta into a inverse because a inverse into Y is again a inverse identity Matrix is just an identity so if you multiply any Matrix with identity you will get the same Matrix so and this gives you a joint of a and this is equal to Delta into a inverse so from this equation you will end up with one formula that a inverse is equal to 1 by Delta into a joint of a and this is the fundamental equation which is known to you from your basic schooling but you never know what is why this equation is coming and many times they tell you for a inverse to exist this data should not be equal to 0 this is the reason this is how this is coming and if this inverse has to exist then definitely this Delta should not be zero because you know definition by 0 is not defined so that's why this ah you know this uh you for a inverse to exist you can say for a inverse to exist this debt a should not be equal to zero clear so this means if you see uh you we generally name this with some technical term a is should be a must be non-singular non singular Matrix you can say a must be non-singular Matrix okay so this capital A must be non-singular Matrix okay and what's the meaning of Singularity basically Singularity means determinant is equal to zero anyhow okay and here I would like to add one more point it is denoted as you can say it is denoted as we can write here it is denoted as a debt a or sometimes you write this d e t of a both are same both they mean determinant is the actual thing that they are asking you for okay so now if you see huh fine so this is the fundamental equation that a inverse is equal to 1 by Delta times of a joint a is one very important uh you know uh fundamental equation for calculating inverse of the matrices okay so so we have seen that this fundamental equation a into a joint of a is equal to Delta into I is the most fundamental equation that gives rise for this a inverse Matrix now let us try to look at some properties of this determinants okay so if you see properties of determinants properties of determinants because this is one of the very important part where General equations turn over in Gate examination so properties of determinants properties of determinants now let us see certain properties okay so if or straight away first property you can write the determinant of the determinant of the determinant of an upper triangular the determinant of a you can write a here and that's not the oh I think there is some issue in just a minute oh uh just a minute so wait for the program to respond so uh yeah so I hope things are going fine you are getting enough time to make your notes I'm not coming up with a PPT because you know when I write you'll also have some time to make your own notes okay and this video says definitely is going to get all the concepts of your match in very less time so hardly if you spend some four to five hours every day for some seven to eight days let us say you will be very clear to the mathematics portion okay so that helps you in getting very good marks in git obviously so let me have let us try to continue here so of a upper triangular Matrix of a upper triangular Matrix or a lower triangular Matrix a lower triangular Matrix a lower triangular Matrix or a diagonal matrix a diagonal Matrix a diagonal matrix is the product of is the product of principal diagonal elements of the Matrix so this is one very very important property so principle diagonal elements of the Matrix principle diagonal elements of the Matrix okay of The Matrix so this is one really very important a property because many times what they give you they'll give you matrices of high orders four by four five by five and all so when they give you such orders this normal process of avoiding the determinant will not work you know that effective so in such cases you can use this property so directly if you multiply the product of the principle diagonal elements you will get the determinant value so this first property is really some very good you know property in matrices and second I would like to give you the if a is an N by n Matrix if a is a n by n Matrix then then you can write then first then first debt of a transpose will be same as determinant of a so basically if you calculate the determinant of a matrix or its transpose both the values turns out to be the same thing okay and second K into a whole power and or you can say that of K A for any scalar okay for any scalar okay debt of K A is equal to K power n times of this data look basically you might have known this property let us suppose if I have this determinant a 1 1 a 1 2 b 1 1 b or you can say a two one basically a two one and a22 let us suppose if this is equal to determinant of a now let us suppose if I want to calculate determinant of one particular Matrix like for example that of K 1 a 1 1 K 1 a 1 2 means the first elements are multiplied by a constant K1 and second row elements are multiplied by constant K2 so in such case if I want to calculate the determinant of this Matrix you can see this K 1 can be taken out from this first row so K 1 can be taken out from this first half similarly from this second row you can take a K2 out so K 1 into K2 times of this debt a 1 1 a 1 2 a 2 1 a 2 2 basically here okay so now this a one one a one to this determinant of two by two is again nothing but K 1 into K2 times of that a here okay now if you see carefully when a matrix is multiplied by a constant what happens this K 1 k 2 values will be same okay because every element of the Matrix is multiplied by some constant so let us suppose if K 1 is equal to K2 this becomes K Square times that a because it's a two by two so if the Matrix is of order n by n then definitely you will have K power n times of determinant of a okay now we'll go for some other properties also third property you can say the determinant of the determinant of ematics the determinant of a matrix is zero is 0 if one of it so if all the elements if all the elements if all the elements if all the elements in Arrow or column if all the elements in Arrow are column are zeros basically okay so let us suppose if you have a particular determinant in which one complete o is zero or maybe one complete column is zero then definitely if you take the product of cofactors with respect to that then you know obviously zero into some cofactor of that element plus again 0 into something plus again zero into something so this total will give you a zero value that's the reason why in determinants if one o or one column is completely zero then the determinant value is going to be 0 here okay now example you can see I I'll give you one example also better so if you see determinant of this about 3 by 2 minus 1 by 2 0 okay so again Pi by 4 2 by 8 or 2 by long two zero then again if you see for example some values some random values minus one by o two okay E power 4 0. so let us suppose if I give you some determinant like this since this one complete column is 0 this determinant value is definitely 0 irrespective of this element so sometimes they can ask you these kind of things okay so you should be little careful and fourth property you know fourth property if two rows if two rows or two columns if two o's or two columns of a determinant are interchanged of a determinant of a determinant are interchanged if these are interchanged interchanged then the determinant changes its sign then the determinant changes its sign changes its sign so basically what I am telling here is let us suppose you have this determinant a11 a12 a13 a21 g a two two a two three similarly a31 a32 and a33 so let us suppose if you have this determinant now this determinant will be equal to the same determinant value when you swap towards or two columns they come say minus sign means let us suppose if I swap second and third for example or second and third column anything I am swapping second and third row so a 1 1 a 1 2 a 1 3 knob this a21 A2 a23 instead of that I need to give the third O So a31 a32 a double three and a two one so if you take this determinant the difference between these two determinants is nothing but we have swapped second and third OS so in such cases the result changes with a minus sign actually okay so this determinant value will be same as negative of this determinant value clear so when you Interchange to us or two columns the determinant changes its sign this is again one very uh you know important property here now coming to the fifth the value of a determinant of a matrix the value of determinant of a matrix the value of determinant of a matrix uh you know basically the value of determinant of a matrix is 0 or you can say it means unchanged better so yeah it means unchanged it means unchanged if remains unchanged if o or column transformations oh a column Advanced formations Arrow or column transformations are performed on The Matrix are performed on The Matrix are performed on The Matrix I have performed on The Matrix so basically what do I mean by this means let us suppose if I have this Matrix or this determinant for example look let us suppose if I take this determinant a one one a one two a one three a two one a two two a two three so this determinant value will be same as let us suppose if you perform some over Transformations for example you have subtracted second row from first and added the third row let us suppose if you do any Transformations this determinant value doesn't change okay so for example example what I am doing for example what I am doing here example transformation what I am doing is R1 changes to R1 minus R2 plus R3 let us say for example okay so we have done some over transformation so this first row changes as first o minus second plus third okay so which means first element is here is going to be a one one minus a 1 2 plus a 1 3 next this element is going to be a12 sorry I think here we have done some error so a one one minus a21 plus a31 okay similarly here A 1 2 minus A2 plus a 2 3 plus a two three okay and next uh anyhow I am writing this on the other side so you know basically this is 3 2 so this is it just a minute yeah so now let us continue plus a 3 2. and third third element first or third column element will be a one three minus a two three plus a33 so this is the first or First Column element here okay so this is the first row First Column element first or second column and first or third column and the transformation we have applied here is R1 changes to R1 minus R2 plus R3 so this is the transformation we have applied so we are not applying any Transformations on second and third rows so they remain as it is a two one a two two a two three a three one foreign so this is how we get this determinant and if you see both these determinant values will remain the same not only this even if you apply some more transformation C2 changes to C2 plus C3 you can apply any number of Transformations so yes or Transformations are column Transformations cannot impact the value of the determinant okay and I would like to give one more Point again six if twos or two columns if those are two columns or two columns of ematics of a determinant I can say of determinant of a determinant are equal are equal or proportional you can say ah proportional or proportional then the determinant value is 0. then the determinant value is zero okay so then the determinant value is 0 means let us suppose if you have something like this a 1 1 a 1 2 a 1 3 a 2 1 a 2 2 A 2 3 a 2 1 a 2 2 A 2 3 for example then in such case if you try to calculate the determinant value you will definitely end up with a zero sign the reason being these two these two rows are identical actually here correct similarly if two columns are identical also then also your determant is zero or sometimes what happens they can be proportional means let us suppose here have some constant K 1 into a two one k two into a to two uh K 1 into a two one k one into a two to K 1 into a two three in such cases what you will do you will take out the K1 constant and again these two o's will become same okay so that's how if two rows or two columns of hematics are either equal or proportional then definitely their determinant value is going to be zero okay so this is again one of the very nice properties because many times we try to bring determinants in such form so that to make them uh you know to show that the value of the determinant is zero okay so that's why this is one very important property again and now seventh Okay so just a minute yeah so seventh property determinant of a into B is basically equal to product of the individual determinants that a into that b okay now I'd like to give we have talked about a lot of special type of matrices orthogonal Matrix you know we have talked about determinants of lower triangular upper triangular diagonal matrix so now let us try to talk about this uh you know uh determinants of this special matrices that they told you orthogonal Matrix in value add important all these cases basically okay so let us see what are the cases eight the determinant of the determinant of the determinant of SQ symmetric Matrix the determinant of SQ symmetric Matrix of odd order of odd order what's the meaning of odd order you know if you have a matrix of this size n by n because it is Q symmetric so it has to be square Matrix n is odd okay n is odd basically so this is SQ symmetric Matrix of odd order is always zero is always zero how this property is one really important property that determinant of SQ symmetric Matrix of odd order is also zero essentially in j e Advanced also you have a question on this basically in this year so anyhow let us see how this is happening means if you take any skill symmetic Matrix of odd order third order fifth order seventh order ninth any any order so that value is always equal to zero how come see here so I would like to give you this analysis here you can see a is Q symmetric correct so here this a is SQ symmetric Matrix so a is a skew symmetric Matrix here correct so if a is Q symmetric Matrix then a is equal to minus a transpose which you know now if you apply determinant on both matrices because both these matrices are same minus a transpose so this implies Delta is equal to look to this a transpose can I take this minus 1 as some scalar multiplication because K value can be minus 1 also so if you take this minus 1 as multiplication to this a transpose so if I want to take this minus 1 out of this determinant I know I need to take out this K as K power n times so let us suppose if the Matrix order is of n by N I can write this as minus 1 whole power n into that of a transpose so from this equation what can we write data is equal to minus 1 whole power n into you know this property determinant of a transpose is same as that a so this is again equal to that a so from these two equations you can understand one thing this implies that a minus minus 1 whole power n times of Delta is equal to 0 and this if you take that t a common you have 1 minus minus 1 whole power n is equal to zero this is the equation that you attend with but now try to observe one thing let us suppose if n is even for example okay if n is even then what happens this minus 1 whole power n gives you an even number correct so if n is an even number minus 1 whole power n gives you plus 1 and 1 minus 1 is always 0 and this equation gets satisfied irrespective of this data being 0 or not correct I am repeating if this n value is even then this minus 1 whole power n will become 1 and this 1 minus 1 product will definitely come out to 0 correct so even if that is 0 or non-zero doesn't matter this equation always holds good when n is even so that's the reason when n is even you can't comment anything now if n is odd if n is odd if n is odd then you know minus 1 whole power n is equal to minus 1 correct so this implies that year times of 1 minus of minus 1 which is equal to 2 times of Delta is equal to 0 and this gives you depth of a is equal to 0 okay so this data is equal to zero under two conditions first thing a is qismatic Matrix and more more importantly this n is odd number correct so that's why if you take any determinant of Q symmetric Matrix of odd order definitely the answer turns out to be zero clear so many times they ask you they give you questions like this in exams the determinant of the value the determinant of The Matrix the determinant of The Matrix the determinant of The Matrix given by given by they give this Matrix like this a is equal to a i j for all one less than or equal to I comma J less than or equal to 5 it is a five by five Matrix and a i j is equal to a i j is equal to I Square minus J Square for example okay so they have given you some complicated function I mean it's not very complicated but again slightly complicated so I Square minus J Square means if you want to calculate third or fourth column element 3 Square minus 4 square is going to be a34 correct so like this if you try to frame a matrix it takes lot of time to form the Matrix second thing it's a 5x5 Matrix so even after forming the Matrix the time taken by you to calculate the determinant will be very large so that's why if You observe one thing carefully in such cases a i j is equal to I Square minus J Square correct so if I calculate a j i what is this J Square minus I Square obviously that so if I pull out a minus sign this is I Square minus J Square which is nothing but minus this I Square minus J Square I can replace with a i j so that's why if you see this Matrix a j i is equal to minus a i j this is the notation for this skew symmetric Matrix okay so if this is the condition then this Matrix is skill symmetric Q symmetric Matrix so obviously you know that this Matrix is Q symmetic Matrix and N value is odd so definitely this determinant is going to be zero okay so n is equal to 5 n is equal to 5 which is odd so therefore you can decide directly that that a is equal to 0 okay so I am telling you in general here it need not be two it can be 3 4 5 anything any number as long as it is of the format I power K minus J power K definitely it's going to be SQ symmetric Matrix because of this two three steps what I have written here clear so that's how the determinant of every odd order skill symmetic Matrix is basically zero clear I hope I am clear till this point so let's go to the next property next important property so the determinant of the determinant of an orthogonal Matrix that what the determinant of any orthogonal Matrix you can say the determinant of any orthogonal Matrix is plus or minus 1 that's it okay you can never find a orthogonal matrix being singular which means you can never find any orthogonal Matrix whose determinant is zero basically clear because orthogonal Matrix can take only two values for its determinant one is plus one another is minus one okay let's see how this happens look you have a times a transposed is equal to I correct so I a transpose into a is also equal to I that is one very important thing so now if you see if you apply transpose to both sides of this I mean so if you apply determinant to both sides of this equation this is product of two matrices so I can write them as individual products that a into debt of a transpose is equal to we all know the determinant of identity Matrix is one so this implies that a into now determinant of a transpose is same as that a so this is again that a and this is equal to one so that a whole square is equal to 1 so this determinant of a is basically equal to plus or minus 1. so if you find one thing I have told you in a status class also normally since you know what is inverse and all I'll tell you one thing I told you a inverse is equal to a transpose for this Matrix correct if you remember for this orthogonal Matrix a inverse is equal to a transpose now for the given orthogonal Matrix you can always always have transpose because transpose is nothing but just interchanging rows and columns so if a transpose is always present a inverse is also always present and you can clearly know from this this determinant is never going to be zero so there is no obstruction for the formation of this a inverse clear and basically I want to tell you one thing what is orthogonal Matrix also I'll tell you basically so if you take any two arrows on any two columns of any orthogonal Matrix for example I have given you this Matrix cos Theta minus sine Theta zero sine Theta cos Theta 0 0 1 so basically if you take any to those or two columns then those twos and two columns will be perpendicular to each other what's the meaning of this let us suppose if I write this as cos Theta I bar plus minus sine Theta J Bar plus 0 into K Bar similarly sine Theta I bar plus cos Theta into J Bar plus 0 into K Bar means if I take these three hours as three vectors for example then if I take any two vectors then definitely the dot product of these two is going to be 0 you can tie if you want look If I multiply these two rows for example cos Theta into sin Theta sine Theta cos theta plus minus sine Theta into cos Theta this 2 gets canceled 0 into 0 0 again similarly if you take any tools zero into sine Theta 0 into cos Theta 1 into 0 again 0. so dot product of any two Vector any two uh you know a rows if you take by General this or columns also it can be for example you can take first and last column any two columns this dot product will always MN 0 which means and that's the name why this Matrix is also called orthogonal means perpendicularity if you take any two those are columns and if you do the dot product of these vectors definitely you will get 0 here clear so that's why this analysis and this Matrix is one very special kind of Matrix actually clear now let us go for some more properties the determinant of the determinant of an item potent Matrix an item potent Matrix foreign is either 0 or 1 again how we know this again look we have seen the definition of idempotent Matrix so a square is equal to I so if you apply determinant on both sides Delta whole square is equal to I that I so this that year whole Square can be written as you can write Delta Square this can be written as that a whole Square which is equal to 1 so that a is going to be I'm sorry so we are talking about item button right here then this is not I this is a because item potent is a so if you see this is Delta and this is that a so if you bring this that t s Square minus that a is equal to zero so this implies if you take the t a common then you have Delta minus 1 is equal to zero so this is a scalar multiplication so definitely if the term has to be 0 this implies either that year should be 0 or data minus 1 should be 0 which means that a can take two values which are nothing but either 0 or 1 so it's not always guaranteed that item potent Matrix have an inverse because probably the item potent Matrix can have a zero determinant and obviously you may not find the uh you know inverse sometimes okay so whereas in case of orthogonal you can always assure there is some inverse existing okay so anyhow this is the case now let's talk about involutely also the determinant of the determinant of an involutely Matrix the determinant of an invaluative Matrix is again plus or minus 1 okay as you have seen just now so a square is equal to I so this implies that of a square is equal to that of I so this implies that a whole square is equal to this one which means that a is equal to plus or minus 1. so this is again one very important uh you know uh result again okay so sometimes they ask you in examination they will give you choose the wrong statement the determinant of an invaluative mat x is 0 such sentences they can give you okay so at that time you should know what is invaluative why these things are happening but trust me one thing I am telling you your University exam maths is much lengthier as compared to gate maps and gate match is relatively very simple clear so now let us try to understand this fine this formulas and all are fine but uh let us suppose if you multiply two matters because next property I'm going to talk about some something about determinant of multiplication something like that now try to understand one thing let's suppose you're actually multiplying some matrices okay for example two matrices then what actually you are doing okay this means okay multiplication wise you know this into this you will multiply additions everything you'll get one Matrix but actually what is that multiplication doing okay so basically these are called transformation of matrices or you can say this called transformation matrices means let us suppose if I have for example I'll give you one simple example let us suppose you have some plane for example okay so this is plane this is plane now let us suppose we have a line Y is equal to X which is passing through the origin of course Y is equal to X Y is equal to X this is x axis this is minus X this is y this is minus y and let us suppose this is the line Y is equal to X now let us suppose you have taken a point a comma B for example here a comma B now let us suppose this point if you if this you know coordinates if you write as a matrix form a B for example a b let's suppose you have written like this now try to understand one thing or your exercise a B for example like this in the vertical space a B like this right this point now let us suppose if you want to find a matrix which when you multiply to this Matrix let us Suppose there is some Matrix a for example if you multiply this Matrix a to this Matrix let us suppose your intention is to find the reflection of this point to find the reflection about this line Y is equal to X okay so you want to find this Reflection Point and you know this Reflection Point is going to be B comma a for example because if you take midpoint and all but uh you know uh product of slopes and all you will get this point become a very easily fine okay now what Matrix a when you multiply with this a comma B will transform this point as a reflection of this given point means basically if you see if you multiply a matrix a to this you will definitely get let us suppose this is size 2 by 2 this is size two by one you will definitely get two by one Matrix here this much you know but what Matrix a can make this coordinates a comma B to get swapped means b e a so these matrices this matrix multiplication is actually in geometrics and what it's doing it's basically reflecting the point now what is this point what is this Matrix a let us try to understand okay let a is equal to let a is equal to a one one a one two a two one a two two it's a 2d space so I am taking two white now if you multiply this Matrix with a B let us multiply a one one a one two a two one a two two this if you multiply with this Matrix a b you are going to get ba correct so let us see what is this elements so if you multiply a times of a 1 1 plus b times of a12 similarly a times of a two one plus b times of a two two this Matrix is going to be B A now you know if two matrices are equal the corresponding elements must be same so now if you equate these two elements this first element with this first element you can easily understand a one one value is 0 and a 1 2 value is 1. similarly if you equate this element with this element you will come to know a two one is equal to 1 and a22 is equal to zero so if you form this if you replace these values in this Matrix you will get a is equal to 0 1 1 0 so basically this is the Matrix which can transform this point means you can see this point can be transformed as Its Reflection if this coordinates are multiplied with this particular Matrix which is given by 0 1 1 0 so this is how matrix multiplication is actually transformation of some given matrices with respect to the the your interest your point of interest you see something I have I might have done is a reflection about x axis or something like the reflection about Y axis I would have done anything okay so these coordinates this matrices decide how these points gets transformed similarly you have uh you you know in plus one or plus two you might have gone to something called transformation of axis okay so this cos Theta minus sign Theta sine Theta cos Theta if you see if you multiply that Matrix to a point you will get some other values of X comma y so if you evolve the axis by some Theta degrees then the new coordinates are given by this transformation matrix cos Theta minus sine Theta sine Theta cos Theta you know you have learned these things this is the actual meaning of multiplications now why I have talked about this because again next property I'm going to talk about this multiplication uh of matrices look if product of two non-zero matrices if product of two non-zero matrices if product of two non-zero matrices A and B is 0 is 0 Matrix basically is a null Matrix you can write is a null Matrix is a null Matrix if product of two non-zero matrices A and B is a null Matrix for example then what happens basically that a is equal to 0 and that way should also be equal to zero look normally I'll tell you one thing many of you may feel that if a and b are non zero matrices but let us suppose if a b is equal to 0 whenever a b is equal to 0 if you apply determinant on both sides you know that that I into that b is equal to zero so by scalar equation you may feel that if one of these two if data is zero or if that b is equal to 0 then this equation holds good but actually if some case like this happens then both you can write then both should be singular matrices both matrices should be singular both matrices should be both matrices should be singular okay so this is one very important Point why I will show you now okay because in it we have a previous year get question also like this they'll give you that a equal to 0 that will be not equal to zero that a is not equal to zero that'd be equal to such options and our intuition tells that one if one of them is zero that's good enough but actually both has to be 0 why I will tell you now look we'll delete this now see here why both has to be zero look let a is equal to 0 and that be not equal to 0 for example okay let us suppose this happened one of this is zero so that again a into B we have 0 so this immediately tells you that b inverse exist B inverse exist that means if that a is equal to 0 and that b not equal to 0 then definitely you know B inverse exists so if you take this a b is equal to 0 and if you multiply with this B inverse then you will get one thing this B into B inverse is I so a into I is equal to 0 mat X because 0 into something it is 0 this is given which is a is equal to 0 which is actually wrong as per our assumption that two non-zero matrices so A and B should be non-zero matrices but if you feel that that b is not equal to 0 you are trying to understand that a is zero which is actually contradiction to a uh take intake that both A and B are actually uh you know a non-zero matrices so that's why similarly if you take Delta not equal to 0 and that b equal to zero you will find out b equal to 0 okay so therefore foreign should be equal to zero okay so this is one very very important property again in uh determinants basically because many times they you know they keep some questions around this for you so that's the reason why it's one very important property the product of two non-zero matrices A and B if it's a zero Matrix then both the matrixes should be definitely singular matrices okay both the determinant should be zero clear so please make a note of these things very soon when we come to the practice question session or something like that it will be very helpful for you okay so and majority you can you may be having some p y cues with you by this time so once attending this lecture or four to five hours almost you can definitely solve almost some 95 percent of the questions in engineering maths I mean in linear algebra okay don't worry remaining five percent I'll solve in the pyq sessions okay fine so let us continue let us continue here so I would like to talk about one very important uh you know uh value actually value of one determinant at this point moment okay so I'll also give you some technique uh for writing okay the things if any writing anything by the determinant easily okay look so see here let us suppose if a is a and by n Matrix n by n Matrix with determinant with determinant as DTA with the determinant as that a for example let us suppose if this is that a then of adjoint of a is how much so many times what they do they don't give you the actual Matrix if they don't give you axial Matrix then you cannot calculate that joint but they'll ask you about that of a joint a and this question is one very famous question in you know get domain so let us see here if a is an Matrix with determinant as that a then what is the determinant of a joint of your basically look I'll tell you we have this equation that fundamental equation that a into adjoint of a is equal to Delta into I that you have this equation now if you apply determinant on both sides depth of a into a joint of a is equal to debt of data times I now try to understand one thing this is product of two matrices so I can split it that a times debt of a joint a is equal to now this data is again some number so and if a is of size n by n this I is also of size n by n so if you want to take this Delta out as a constant out of this determinant then you know since this I is n by n Matrix this turns out to be that year whole power n times determinant of I now you know one thing this determinant of I value is actually one so you can cancel that a some n minus 1 times get a power n minus 1. so therefore you can understand one thing that of a joint a is how much that year whole power n minus 1 so this result is one very very important result for gate examination okay many times you see you can if you will find a question from this uh you know from this actual equation okay sometimes they go one step ahead and they'll put one more adjoint here at joint off at joint here they'll ask let us see what is the case let B is equal to at joint a for example for example you have you have a matrix a you have calculated its adjoint so let us consider that Matrix as B now since this B is also a matrix we can write B into adjoint B is nothing but debt B into I I can write this correct so if you again simplify you will get that of a joint B is equal to debt B whole power n minus 1 you will get again the same analysis if you follow you will get that of adjoint B is that B whole power n minus 1. now if you replace b as a joint a so this implies debt of a joint of a joint a if you do this then you will get something like this that b is nothing but debt of a joint a which you have that here whole power n minus 1 here so dirtier whole power n minus 1 this whole power n minus 1 okay so if you simplify this you will end up with something like this a joint of a joint a is equal to that a whole to the power n minus 1 whole Square okay so this is how we have this relation now try to tell me one thing let us suppose if you have taken one Edge joint you got n minus 1 n minus 1 whole to the power 1 for example you can say when you took two head joints here there is two n minus 1 whole Square came now let us suppose if you put one more joint this turns out to be Cube It's a very simple analysis again like the same thing if you follow so many times without giving the actual Matrix and even if they give actual Matrix it takes you so much time to calculate this adjoint a okay so that's the reason if you have this uh determinant of a value you can easily tell determinant of a joint a once you know the size of the Matrix means basically once you know the N value you can uh you know keep going ahead for the determinants of the head joints clear so these are very much important properties in uh determinants that I want to talk about at this moment okay so I think I have discussed some nine to ten properties of determinants which is again very uh you know good thing so this determinants are very important uh basically for your understanding okay so we have seen the determinants of all the types basically here clear now one thing uh fine so this is this with this we close this second uh part of this video which is determinants okay so we have seen what is the determinant we have seen determinant is nothing but the summation of product of elements of a column with the corresponding cofactors okay so we have seen that and we have seen certain properties of determinants which are very you know useful in general uh examples outside now let us try to Define one ah the next part of this so system of equations correct so the main process of learning this linear algebra solving is basically to solve simultaneous system of equations okay but to solve this simultaneous system of equations or whether you can say even before solving many of you should know like whether if I give you some equations whether this equations will have a solution or not look for example if I give you two equations I am just explaining so let us suppose if I give you two x two equations 2x plus 4y is equal to sum of 8 and again X Plus 2y is equal to 9. let us suppose if I give you these two equations these are two equations for you that's it but actually without solving you can tell whether this equation okay if you try to solve you will end up with this equations do not have any solution but normally it's a two by two case it's a simple case but let us suppose if I give you some big case four by four five by five such cases you should know some techniques some mathematical checks to understand whether the given system has a solution or not okay so that's how how we check that so for checking that we will be in need of one very very important concept which is called rank of a matrix okay so this rank of a matrix is one very important concept and we'll see the complete system of equations deal 7 with this rank of the Matrix okay so let us see how we go ahead here so let us put a heading a rank of a matrix a rank of a Matrix Matrix so what is this rank of a matrix okay so this is one very important concept and we all should listen carefully here okay so for next 10 minutes please don't uh you know uh look anywhere be focused so rank of a matrix so a real number a real positive number you can say a real number in general you can say r is said to be a rank of a matrix a rank of a matrix a m by n okay so this area number are smaller you can say is said to be Bank of a matrix a m by n here you need to observe one thing this Matrix need not be a square Matrix to talk about a rank okay for example if you want to talk about determinants then definitely should have a square Matrix for rectangular matrices you cannot talk about determinants but if your idea is to talk about this a m by n i mean if your ideas to talk about this rank of a matrix then your Matrix need not be a uh you know basically a square Matrix okay so now see real number R is said to be rank of a matrix a m by n if first point first point all the minus all the minus of size R plus 1 by R Plus 1. and above and above are zeros as second there exist at least one non-zero minor non-zero minor of size R by R okay so what is the meaning of this okay I have written two conditions here so first of all all the minus of size more than R are definitely zeros and they exist at least one non-zero minor of this uh you know Matrix a m by n now let us see how we practically do this okay look let us suppose I have taken some example because examples make you understand better so let us suppose I have taken some example so one two three four let us suppose I have taken these elements two four six eight then uh say three five seven nine let us suppose I have taken now try to understand one thing this is one Matrix this is of size three by four for example I have taken some random Matrix okay so this is some Matrix basically and it's of size three by four now how do I find the rank of the Matrix okay so let us check here let us check here so if I want to find the bank of this Matrix first of all I need to see what is the highest non-zero minor that is possible so minor is basically some determinant some part of the Matrix correct so tell me if this is a three by four Matrix what is the maximum size determinants I can take out of this Matrix can I take four by four because four is the highest no because there are only three rows in this Matrix so highest order that's possible is third order minus third order minus okay we have seen this third order minus so what are the third order minus that you can take out of this Matrix one two three two four six three five seven means basically I have deleted the fourth column now I can take one more possible minor what is that let us suppose I'm deleting the third column now one two four two four eight and three five nine now let us suppose I have one more minor if I delete the second row one three four I mean second column 134 268 and uh three seven nine and let us suppose I am deleting the first column two three four six eight five seven nine so these are the four three by three minus that you can form using this three by four Matrix correct now let us suppose if the if one of these determinants is not equal to zero for example let us suppose even three are zero but at least one determinant is non-zero then we declare the rank of this Matrix is 3 because we cannot form four by four matrices or above uh means we cannot form any four by four determinant so above determinants and third order determinants are the highest order determinants that we can form and if you find at least one non-zero minor then definitely you can declare rank of this Matrix is three but let us suppose if all these four are zeros now we need to go for Next Level so third order we have seen means 3 by 3 we have seen now if all these are zeros then we'll go for second order minus okay so second order minus second order minus so basically two by two okay so 2 by 2. so now let us try to understand one thing two by two minus but first before going to 2 by 2 let us calculate these minus so undoubtedly you can see this determinant value is 0 why because first row and second are proportional to each other correct second row is just two times of the first row so if you take one two out first one second row become identical same is the case here 0 same is the case here also 0 and same is the case here also zero so basically here all the four determinants are zeros so we cannot find any third order minor or this three by three size minor as non-zero so then we need to go for second order minus again if you want to avoid second order minutes you'll get lot of minus here correct you can take one two two four two threes four six one three four one three two six again you can get a lot of minus but at least there should be one non-zero two by two minor so let us suppose I'm taking this minor for example four six five seven so if you see this value this is not equal to zero because five sixes are thirty four sevens are Twenty Eight these two are not equal so you can form any number of two by two minus but at least you are pretty much sure that you have at least one non-zero minor of size two by two so now if you want to declare a rank of this Matrix a rank of a is equal to 2 if you want to declare this then you should make one shot one thing very clear that if you take size higher minus means you know if you want to declare this 2 as the rank of this Matrix a then you should pretty much show that all three by three four by four five by five possible of course all these things are zeros and definitely they exist at least one two by two minor of non-zero value okay this is what I have written actually here so if you want to declare this number smaller as a rank of this Matrix all the minus of size R plus 1 by R plus 1 and above are definitely zeros and at least you should have one non-zero minor of size this R by R if you do the R value with the two you will understand the next example that's it okay now so basically checking this so if you understand one thing many of you might have had this question by now how do you know how many two by two miners you have and how do you know whether in that two by two minus you may have zero or not look I'll tell you if you want to form two by two minus you will get 3 c 2 into 4 c 2 so this is basically 3 into 6 you will have 18 2 by 2 minus totally okay so writing this 18 2 by 2 minus and checking whether they are zeros or not is not very easy here right now and of course if the Matrix size is of some six by six seven by seven of such kind then forming four by four minus five by five minus this is not very uh you know very efficient way to check what is the rank of a matrix so what we have done for this is you definitely should have some easy procedure or some constructive way to decide what is the rank of a matrix so now what we have done is we talked about one thing which is called row equion form of a matrix this is one very very important part row eclion form of a Matrix Matrix now before going for working on form I forgot to tell you one uh small thing here look what I want to tell you here is it is denoted by you can write here it is denoted as it is denoted by you can say a row of a is equal to R okay so this rank of a is actually related with the symbol oh that's common symbol you take for resistivity or density in mechanical and civil so if you see this is the symbol that you denote for this rank so rank of a is denoted with this row of is equal to R okay so now let us come to this however client form okay so let us come to this very clear form of a matrix how do we know that uh basically what is this over equivalent form let us see then we will see if a matrix is in the working form or not we'll also see how to convert any given Matrix into a vehicle iron form okay we'll see these things look a matrix let me change the color a matrix a matrix capital a m by n is said to be is said to be in a row a Cleon form is said to be in a row equion form if if again there are two conditions here if first condition number of zeros before the first non-zero element before the first non-zero element before the first non-zero element in Arrow before the first non-zero element in a row basically so if you take any Matrix for example then in a oh if you take definitely some elements can be zeros and some can be non-zeros so if you take the first non-zero element that's coming if you count the number of zeros before that so this should be foreign should be more than should be more than number of such zeros such zeros in its preceding space reading means one above that units preceding go preceding row and second thing second thing number of years before the first non-zero element in a row should be more than number of zeros in its precedingo and importantly we can have second thing as 0 goes if any means let us suppose during some Transformations if you get any zero rows for example then should lie at bottom of the Matrix should lie at the bottom of the Matrix should lie at the bottom of the Matrix should lie at the bottom of the Matrix means basically let us let me show you one example ah in our equivalent form you will understand that okay so let us suppose if you take this Matrix which is in this form for example uh one two three four five six seven eight all zeros for example let us suppose you have taken it need not be square Matrix also it can be some other Matrix as well no issue now what is the meaning of the first statement that I have written here this Matrix is actually in a weekly on form this is in no equivalent form how I'll tell you okay on form oh a clone form now how actually this uh this two points will actually in line with this example that I have given you here let us check so first statement number of zeros before the first non-zero element in Arrow should be more than number of such zeros in its preceding look what's the meaning of this let us suppose I have taken third for example if I take this third o if I take this third row then this is the first non-zero element in this row correct so this is the first non-zero element and number of zeros before the first non-zero element R2 in this case now this number of zeros before the first non-j element is 2 so definitely in its preceding o you should have one either or zero elements so basically here you can see one why I have intentionally kept one here because if you take the second row this is the first non-zero element so here if you have one zero here there should be no zero that's the reason why I have taken like this so if you take any particular row then the number of zeros before the first non-zero element should be definitely more than number of such zeros such means again if you take this second row this is the first non-zero element and these are the zeros before this first non-zero element so if you see this number of zeros before the first non-zero element should be definitely more than number of zeros in its speciating so for third number of zeros should be more than the second row for second row number of zeros should be more than the first row like this okay so so this is what I mean second point it's a easy point you can understand zeros means basically if there is some any error which is in which all the elements are zeros they should lie at the bottom of the Matrix basically so if you see this one zero which is completely zero so this should lie at the bottom of the Matrix actually clear so that's how you know this Matrix is in row equilion form okay now basically okay I have given one direct example and I made you understand how aortically on form of Matrix look like but let us suppose if I actually give you one Matrix then how you convert that Matrix into a equilibrium form that's the very important thing okay so that is what next I am going to discuss that conversion of conversion of conversion of a given Matrix a given Matrix into a row a clion form this is one very important part that you should know basically here okay so conversion of a given Matrix into conversion of a given Matrix into rho equilion form so how will convert look let us suppose I have given you one Matrix let a is equal to a 1 1 a 1 2 a 1 3 A 1 4 let us select 3 by 4 a two one a two two a two three a two four a three one day three two a double three and a 3 4 for example so let us suppose this is the Matrix now how to make this elements uh make this Matrix a uh you know into a clear one form let us see first of all try to make these two elements zeros first of all try to make these two elements zeros Okay so when you try to make what are the Transformations you'll apply are two changes to R2 minus R2 minus a21 by a 1 1. into R1 and R3 changes to R3 minus a31 by a 1 1 times R1 so let us suppose this is the transformation you have to apply now if you apply this transformation then undoubtedly these 12 minutes will become 0 and all these six elements will change okay now here you can have one question okay sir we'll do this but let us suppose if a11 is 0 and these two are not zeros okay if this element is 0 but these two are non-zeros then how to do this look simple you can swap those then okay so when you swap here you'll get a non-zero and maybe one zero you'll get in uh some cases okay so that's how you actually perform this transformation then this Matrix gets equivalent to first order doesn't change a one two a one t a 1 4 these two turns out to be zeros then you have a two two dash because this value changes after this transformation a two three dash a24 dash similarly a32 Dash foreign form why because if this is the first non-zero element you have one zero so definitely here you should have minimum two zeros but you have again only one zero so that's the reason we will make this element against zero how will you make that again one more transformation R3 changes to R3 minus a dash T2 by a dash 2 2 times R2 this is what you can apply so if you apply this transformation first and second remain as it is so first and second row MN as it is a one one a one two a one a one four then again you can have zero zero secondary elements as it is a two two dash a23 dash a24 dash and this third row it changes this 0 minus something into 0 this will be zero so this is also one more zero here and again these two elements will change a double three Double Dash a34 Double Dash again so this is how you have a basically a equilion form now if you see in this Matrix there are no zeros so there is no nothing at the bottom but here you can see before the first nounselment there is no zero before the first non-z element there is at least one zero it should be definitely because here it's zero it should be more than zero similarly here they have two so this is how you convert any given Matrix into a row equilion form actually so now if it's of size three by five or five by five whatever it is you can always have this conversion and anywhere if you are getting this denominator element zeros then just swap those and try to avoid the zero division that's it okay so now why we have taken so much pain in converting a given Matrix into row equivalent form what is the relation between this row equivalent form and rank of a matrix so if you want to understand this you will get this connection with this formula a rank of a matrix a is equal to number of non-zero rows number of non-zero rows in row equion form of a rho equilion form of a basically here okay so if you take a rank of a that is basically equal to number of non-zero rows okay so once independent of the size of the given Matrix if you can convert The Matrix into uh you know over equivalent form then definitely can identify rank of the Matrix very easily clear so that's a one common thing again so that's why we have taken so much of pain in converting the rank of the Matrix to uh you know by knowing its a weekly on form now one thing that I would like to tell here why we have learned this concept of rank as I told you to talk about system of equations correct so let us without doing any time okay before going for that I would like to give you some properties of this rank of the Matrix which could be you know helpful for you so first property or I can say properties of properties of rank you can write some two three properties which are good so first rank of a m by n will be definitely less than or equal to minimum of M by n m comma n correct so basically we have seen if it's a t by 4 Matrix at maximum line can be three or maybe less than that depending upon the minus being zeros or not okay so out of these two which is minimum the rank of this can be at maximum equal to that or it can be less than that and similarly you know rank of a is always equal to rank of a transpose because the minus don't change even if you take transposes of them okay so that's the reason and third one very good property I would like to give you rank of a into B will be less than or equal to minimum of rank of a comma rank of this Matrix B here okay so that's how if you do the product of two matrices rank of a multiplied with B then this is less than or equal to minimum of rank of a comma uh you know rank of B so this uh basically are some good important properties and especially the third property is somewhat you know uh not very frequent but sometimes it's good to know in Gate examination when you're going for matrices okay so that's how we have this the matrices now three properties now let us go to one very important part of this linear algebra the main core of this system of equations system of equations so almost I am going to spend some 30 minutes of duration on this system of equations so this is a very important portion that you all should actually uh you know put some interest in okay so system of equations let us see what is the system of equations and how it actually helps you out okay so let us deal with the case of the the equations with the unknowns let us start because journalism not very easy not very difficult also to deal with okay so let the given system of equations let the given system of equations be let us suppose you have three equations with the unknowns so let me write them a 1 1 into X plus a 1 2 into y plus a 1 3 times of Z is equal to B1 a21 times of x plus a 2 2 times of Y plus a 2 3 times of Z is equal to B2 a three one times of X plus a 32 times of Y plus a33 times of Z is equal to B3 okay so let us suppose these are the three equations with the unknowns now your job is rather than solving okay you have the techniques to solve we have many methods like Gauss elimination all these things but right now let us suppose if you want to know whether this system of equations has a solution or not okay so means if you try to solve this equation so will you get a solution this is what your interested to know right now so if you see this same system of equations can be written as matrices like this how come I can write this Matrix a11 a12 a13 this is a two three a32 a33 this if you multiply with this Matrix x y z is equal to B1 B2 B3 so this is how you can actually write this system of equations into this matrices how come because if you see this is a size this is of size 3 by 1 so obviously this product is of size three by one now if you check this if you multiply these two matrices on the left hand side a one one into X plus a one two into y plus a 1 3 into Z this is what you'll get as the first element first element second row a21 into X plus a two two into y plus a two three into Z similarly third one so if you see these left hand sides of the equations are nothing but the elements in the matrix multiplication of these two matrices okay so that's how this given system of equations can be you know converted as a product of basically some matrices okay so this can be written as a into X is equal to B so this system of equations can be written in terms of some Matrix equations a x is equal to B where a is called this capital A is called coefficient Matrix coefficient Matrix so why it's called coefficient Matrix because you see clearly this is basically your a and this Matrix is formed by the coefficients of the variables in this equations next this x is called variable Matrix variable Matrix okay so variable Matrix because that this x capital x I am denoting this this capital x is basically here uh you know the Matrix containing all the variables involved in the equations so this is variable Matrix similarly this is B so if you see this B is basically a constant Matrix these are the constants on the right hand side so this is called constant Matrix which many of you have seen correct so now people will tell if you want X then X is equal to a inverse into B you have one method you have comma so you have Gauss Jordan Gauss elimination they have n number of methods okay but my job here right now is not to find the solution but to know whether this equations have a solution or not okay so let us see what how we can do this so ax is equal to B now a system and you can see carefully here this equations are non-homogeneous equations why if you take this equation this is first degree equation first digator because here you have a power one here also you have a power one here also you have a power one so all these three have power 1 but here the power is not equal to one for example if you write this Z power 0 or X power zero anything the degree is zero so all the four terms of this equation do not have a same degree so that's why this is called non homogeneous equations okay so let us write five non homogeneous system fire non-homogeneous system ax is equal to B ax is equal to B first case if rank of a is equal to rank of a slash B we'll see what is this a slash B this a slash B is called the augmented Matrix okay means basically if you write a slash B augmentation means addition Okay so a11 a12 a13 a21 a22 3 B I mean sorry a32 and a33 for example so if you take this then B augmentation augmentation this is called basically augmented Matrix augmented Matrix augmented Matrix for example so augmentation means addition means you will add the column B to this Matrix so B1 B2 BT this is how you will have your augmented Matrix basically the three by four in this case Okay so this is the augmented Matrix so now if you see if this rank of a is equal to rank of a augmented B is equal to number of unknowns in the equations here you have three unknowns basically so number of unknowns if this happens then the system has unique solution means it will have only one solution then the system have other than system has unique solution unique means only one solution unique basically one solution unique means basically one solution now it could be possibly that case if a rank of a is equal to rank of a augmented B but this time both these values are less than number of unknowns less than number of unknowns then the system has I'll take examples of each of these three cases total there are three cases I'll take examples I'll show you geometrically what is happening when you uh when these conditions gets hold okay so then the system has infinitely many solutions infinitely many infinitely many solutions and third if rank of a is not equal to rank of a augmented B I will show you how these three cases can come so if rank of a is not equal to rank of a augmented B then the system has no solution then the system has no then the system has no solution basically here okay then the system has no solution how we can have this kind of systems basically when you have a system with only one solution normally when you solve two equations you will get some solution how a system can have infinitely many solutions how a system can have zero solution let us see the cases of each of them and let us try to understand I hope this is a very long video so I expect you to be patient because obviously it's a you know very short time and forget 2023 so definitely you need to spend some good time on this mathematics because started with 13 marks and unknowingly or indirectly you can say many other marks will be coming from this engineering math subject I have taken this first video to be on linear algebra because you will feel little comfortable and next video we are going to make on basic calculus okay so anyhow let us see here continue so case one let us suppose I give you two equations this one let us suppose I have given you two equations for example 2x minus y is equal to sum 7 let us say and uh for example some another equation 3x plus 2y let us say so 3x plus 2y is uh let us suppose again it is 7 okay so if you take uh for example let me calculate this okay fine so let us suppose 2x minus y is equal to 7 and 3X plus 2y is equal to 7 okay it need not be 7 and 7 here in this case it got happened okay now definitely you know one thing if you form this I'm dealing with two by two because I want to show geometrically also if I want to take then I'll need to talk with planes and all so it will be little complicated so right now I am explaining with two by two so you all will understand now let us suppose if you want to uh you know I like this basically uh shall we make the equation simple a bit let us make this constant simple a bit so this is to be this is 8 let us take for example okay so we I just want to make the equation slightly simple so yeah this is fine now so if you see this equations now let us try to convert this equations into matrices okay so if you see a is equal to coefficient Matrix basically what is the coefficient Matrix 2 minus 1 3 2 B is 3 8 and of course capital x is X Y now if you calculate determinant of a that a is equal to 4 minus of minus 3 which is 7 so which is not equal to zero so therefore rank of a is equal to 2 that is clear now a augmented B is how much a augmented B is 2 minus 1 3 2 3 8 so definitely this is a two by three Matrix so the maximum second order minor you can take out of this Matrix is this second order so if you write 2 by 2 minus two by two minus diagram writing so if you write 2 by 2 minus the first minor will be 2 minus 1 3 minus 2 so which is oh sorry 3 plus 2 okay so this is plus two and obviously you know this is not equal to zero so because that's same as the determinant of a which means one thing you got here is very clear rank of a is same as a rank of a augmented B and of course this is equal to 2 and here 2 is nothing but number of unknowns in this equation correct so now if you see a rank of a and rank of a augmented B both are same and of course these two are equal to the number of unknowns in this equation so therefore the given system has a unique solution foreign system has unique solution the given system has unique solution okay so the given system has a unique solution basically here now try to understand one thing ah number given unique solution now how do I know that it has unique solution so basically if you see geometrically I will talk about so these two are the equations or two linear equations so if you try to plot them you will get straight lines cut let us try to plot these tight lines and let us find at how many points they are intersecting okay so because solution mathematically means geometrically the point where the two lines meet each other correct or they cut each other so now if you try to plot these points this is X and let us suppose this is y for example okay so this is your y origin X minus X y and minus y now try to form this line 2x minus y is equal to 3 so if you write this 2x minus y equal to 3 this equation can be written as y equal to 2 X plus 3 correct so if you see basically if you want to plot that line so let us find the x coordinate so if you want the point on you know uh x axis if you put y equal to 0 x equal to 3 by 2. so this is the one point this is one point and if you put x equal to zero y equal to minus three so the line is something like this so okay let me change the ink so you can understand better so this is one line where okay fine so this is the equation of the line 2x minus y is equal to 3. and next if you want to plot the other line 3x plus 2y is equal to 8 so Y is equal to 4 generally when you do this and X is equal to X is equal to 8 by 3 so x equal to 8 by 3 and Y is equal to 4 means if you take this line it turns out something like this so it comes something like this here so if you see these two lines these are the equations of the X plus two Y is equal to eight and you can clearly find from this Matrix analysis you found that this system of equations will have a unique solution correct and of course if you see the number of points this lines touch or cut each other is only one point so that's how we have a unique solution we don't know what is the value of that but right now your job is to identify whether a given system has a solution or not if you want me to be more colorful I have a tool I generally use this in my I've used this for the first time in my research but right now I can show you this graphing calculator by decimals this is one very good graphing calculator so let us see the equations 2x minus y is equal to 3 so let us see yeah 2 x minus y is equal to 3 okay oh I think I have written something so 2X minus y is equal to 3 so this is first equation you see as I have done the first line this line yellow line I have done you can see that yellow line here also like 2x minus y is equal to 3 so what about the other equation 3x plus 2y equal to eight so let us see 3x plus 2y is equal to eight X plus 2 Y is equal to 8 okay so 3x plus 2y is equal to it you have seen some other line coming like this the Blue Line you can see the colors a red line is for 2x minus y equal to 3 the line that is a din color blue line is for means this blue color is basically for this blue line 3x plus 2 Y is equal to eight so if you see these two are intersecting at this point and of course using this calculator you can just know the solution also if you dot this point two comma 1 is the solution of this equations okay but right now we are not interested in solving we are just interested to find how many solutions does this equation have this is from the geometry and this is from the I mean this is from the geometry and this is from this uh you know uh Matrix analysis okay so that's how you can have some system of equations where you actually have the rank of this coefficient Matrix is equal to rank of augmented Matrix and that total value is equal to now you know number of unknowns so in such cases you will have only one solution and if you try to solve this equation you will get only one solution which is x equal to 2 and y equal to 1. so this is the first statement that I'm talking about if this is the case then the system has unique or one solution okay now let us come about the second Case Case two let us talk about case two case two let us suppose I have given you these equations X Plus 2y is equal to 4 and 2x plus 4y is equal to 8 so let us suppose these are the two equations for example okay and practically you can understand one thing right by looking at the equations itself you can understand one thing because it's two buddy you can because it's two by two you can just tell them by looking but for some high systems it's not very possible okay so let us see here X Plus 2y is equal to 4 and 2x plus 4y is equal to eight so let us suppose if you try to you know solve this using the mat axis so let us write these equations so you know a coefficient Matrix is basically one two two four what about this B constant Matrix 4 8 so I a augmented B is how much a augmented B is equal to one two four two four eight now let us try to understand one thing first of all let us calculate that a if you calculate that a 4 minus 4 which is 0 so this data is actually 0 so 2 by 2 minor of this a becomes 0 now you should go for one by one minus okay so if you go for one by one minus what are the one by one minus totally have four one by one values here one two two four and obviously all these are non-zeros so you can say rank of a is equal to one how you know that rank of a is equal to one because second order minor is coming out to be 0 so and you cannot have any third order fourth order so the only possible is first order if you take any of these four elements at least you have one of these four elements as non-zero so that's why rank of a is equal to one now coming to the bank of a augmented V a rank of a augmented B so let us suppose if you take this is of size 2 by 3 so maximum you can take 2 by 2 minus so if you take 2 by 2 minus of a augmented B a augmented B if you take this Matrix and if you take 2 by 2 minus okay so 2 by 2 minus so if you take this 2 by 2 minus then we have 32 by 2 minus what are the 1 2 2 4 means deleted the third column deleted the second column and later the first column so you will find by strange that all these values are zeros because 4 minus 4 8 minus 8 16 minus 16 all these are zeros so definitely you should go for one by one minus followed by and you see in this six elements there is at least one element which is non-zero so rank of a augmented B is equal to 1. so here you understood one thing rank of a is one rank of a augmented B is also one but these two are not equal to the number of unknowns in this equation correct so if you simplify this I means if you write it in an organized way rank of a is equal to rank of a augmented B but this is less than 2 because number of unknowns is two so as per our understanding the system has infinitely many solutions the system has infinitely many solutions and of course I am explaining you for a two by two case because geometrically I can draw some lines and explain you the system has infinitely many Solutions okay so the system has infinitely many solutions now try to understand carefully here if I plot them again this is your x axis this is your x axis and this is your Y axis for example this is your Y axis so Eisen Eisen oh I'm sorry yeah Eisen X minus X y and minus y so if you take this equations so the first equation X Plus 2y is equal to 4 so if you upload that point basically so at x equal to 0 Y is equal to 2 and at y equal to x equal to 4 so which means the line Looks something like this this is the line so x value should be little high so let us suppose I am putting some line like this okay so X is more compatible so this is the equation of the line uh this is the representation of this line X Plus 2y is equal to X Plus 2y is equal to 4. now mathematically okay you got infinitely many solutions but physically how can you have infinite points of intersection this is possible when the two lines overlap each other correct when two lines are overlapping each other you can understand that at each and every Point both these lines touch each other so they have infinitely many solutions so if you plot this 2x plus 4 over is equal to eight again you will get along the same line you will get along the same line okay okay so I am trying to overlap it slowly fine so if you see I have just shown some defense at least so that you can understand that this equation is 4X or sorry 2X plus 4y is equal to 8 and if you see all these points are nothing but points of intersection here this point this point so you have lot of infinitely many points obviously on a line you have infinitely many points so all these are infinitely you know Solutions basically so infinite solutions infinite solutions okay you know basically if I give you a equation of a line how to draw the line okay so if you want if you identify this point and these two points you can draw a line connecting those two points okay anyhow I'll show you on decimals also X Plus 2y is equal to 4 so the first equation so the first equation x Plus 2 Y is equal to 4 so X Plus 2y is equal to 4 this is the line you can see the green line here and what about the next equation 2x plus 4y equal to eight two x oh sorry 2x plus 4y is equal to 8 so 2x plus 4y is equal to 8 so this is in purple color okay one is in green one is in purple but right now we cannot see the gain line because the purple has overlapped this gain one okay so right now you cannot see you can see this as only one line but actually there are two lines which are involved here okay so that's the reason why if you see this they have infinitely many points of intersection always okay so you can see all this at every point there is a point of intersection clear so that's how this system of equations can have infinitely many solutions when this condition holds Good means rank of the coefficient Matrix is same as a rank of this uh what do you say a joint Matrix but at the same time both of them together are less than the number of unknowns in that equations okay so this is one very very important step that you need to make a note of now let us go for the last case so basically that's the third case that we are talking about this is the second case we have discussed let us come to the third case no solution case no solution case okay so let us see case three also case three let us see what is this case three now try to understand one thing sometimes equations can have no solution also how look let me again take one example and let us show again so X Plus 2y X Plus 2y is equal to 4 for example and let us suppose I have taken the equation same equation with some different constant 2x plus 4y is equal to 9 let us say instead of 8 I have taken nine now you can understand one thing if both lines do not have any solution that means practically speaking they are parallel to each other okay if you check the slopes minus of X coefficient by y coefficient then minus of X coefficient by y coefficient is same for these two lines so that's why you can have these two lines as parallel lines in half let us check the matrix conditions so a is equal to 1 2 2 4 so a augmented B a augmented B is equal to 1 2 4 2 4 9. so you can calculate one thing easily if you take a rank of this Matrix if you take determinant of a this is equal to 0 straight away 4 minus 4 0 so rank of a is equal to 1 definitely clear because if you two by two minus r zeros then you will go for the first order minus and obviously out of all these four elements you can see at least there is one element which is non-zero so rank of a is one now what about the bank of a augmented B let us consider so a augmented B if you consider 2 by 2 minus of this 2 by 2 minus because they are the highest order minus you can consider so 1 2 2 4 this is 1 minus one four two nine this is one more minor and two four four nine this is one more minor now you can understand this thing that this determinant is 0 but this determinant is not equal to zero similarly this determinant is also not equal to zero so out of this three two by two minus you have at least one minor which is not equal to zero so therefore rank of a augmented B is basically 2 in this case whereas Bank of a is only one in this case Okay so that's how rank of a and rank of a augmented B both are not equal in this case so this implies rank of a is not equal to rank of a augmented B so this implies the system has no solution the system has no solution generally so what's the meaning of no solution both these lines will look parallel actually feed on them so if you take x axis and if you take y axis so origin X minus X y minus y so now if you see this 2X x plus 2y we have seen previously the line Looks something like this so this is X Plus 2y is equal to 4 this is the equation of the line X Plus 2y is equal to 4 and similarly if you take some other line of 2x plus 4y is equal to 9 will be something like this okay so 2x plus 4y is equal to 9. so 2x plus 4y is equal to 9 and if you see these two lines are parallel and you cannot find any point of intersection between them so mathematically speaking you don't have any solution okay anyhow I will show these lines on this plot the only difference is I made this 8 as 9 okay so let us see so now if you see when I made this 8 as 9 you can find the two lines and these two lines have parallel lines actually you cannot find any point of intersection at any point because these are parallel lines okay so that's how it it's clear that whenever and of course this is a two by two case so I have explained you with straight lines if I go for three by three I need to draw some planes and also it looks a little clumsy so I haven't taken but the same case holds good for any size Matrix the equations with the unknowns the equations with four unknowns anything basically okay so now that's how we have cleared this third point also third Point okay so basically this point and basically this is this system is a non homogeneous system correct this is a non homogeneous system now let us suppose you have a homogeneous system means your all constants b one b two B three all these are zeros then let us see what is the case okay and before going for that I'll tell you one thing look try to understand one thing in this case you got data not equal to zero correct so try to remember this thing whenever if the determinant of the coefficient Matrix is non-zero you always have a unique solution because if data is not equal to 0 here also you will get one minor which is equal to that a okay so that's why these two will be definitely equal if that a is non-zero okay so if that a is not equal to 0 then system will have single or unique solution okay but if that a is equal to 0 then the case falls into one of these two things okay then the system may have infinitely many solutions or the system may have no solution also it depends on the you know for the analysis but if that a is not equal to zero the system will definitely have a unique solution this point has to be very clear in your mind okay now and obviously you know if that a is not equal to zero your equation x is equal to a inverse B you can calculate directly you will get x value okay single solution you can directly calculate this is what it happens now I'll uh write one thing one word here which is actually uh you know yes fine so a system is said to be consistent or a system of equations a system of equations is said to be consistent a system of equations is said to be consistent if a system of equations is said to be consistent if the system has a solution a system the system has solution the system has solution so basically what is the meaning of this this solutions can be unique or infinitely mean it doesn't matter but as long as the system has a solution it's called consistent system it can be unique solution or it can be infinitely many solutions but as long as the signal system has a solution we call that system as a consistent system and for the system to be consistent that a can be zero or it cannot be equal to zero also like this depending on these cases but one thing is very clear as long as your determinant of a is e is not equal to zero the system always and always has a inverse and that system will have a unique solution clear now let us talk about homogeneous system of equations okay so fire homogeneous system of equations you can write homogeneous homogeneous system of equations homogeneous system of equations means basically your right handed term this B Matrix is actually a null Matrix zero okay so whenever the right hand sides of the equations are 0 you can write 0 into X power 1 0 into y power 1 whatever you want and all the four terms of the equation was single degree so that's why now this system uh is called homogeneous system whenever this coefficient Matrix is I mean this constant Matrix is zero this implies a into X is equal to 0 is the system now try to tell me one thing let us suppose if your a is like this a one one a one two a one a two one a two two a two three one similarly the other two equations now if this is a consistent system now try to tell me one thing a augmented B a augmented B will be a one one a one two a one three zero a two one a two two a two three one a three two a double three zero okay so you will have this system now try to tell me one thing if you are interested to calculate the bank of here okay mathematics you will calculate if you want to calculate a rank of this augmented Matrix how we will calculate look a augmented B is a 3 by 4 Matrix a two one a two two a two three zero a31 8 3 2 8 3 this is again zero now if you see this is of size three by four if this is of size three by four the maximum size minus that you can take out of this is third order okay so if you take three by three minus I'll tell you one thing here one very important point I want to evil here so if you want to take three by three minus what are the Matrix what are the determinants you can have a one one a one two a one there I have deleted the fourth column so this is one determinant okay now if you delete the third column then you have a one one a one two zero a two one a two two zero zero you have deleted the third column next if you delete the second column a one one a one zero a two one a two three zero eight three one double three zero this is one minor and the last minor if you delete the First Column a one two a one three zero a two two a two three zero a three two a three three zero so these are the fourth minus now the rank being three or less than that depends on only one determinant which is nothing but this determinant why because this determinant value is 0 we have a complete zero column okay you have one complete column of zeros so this determinant is zero this determinant is also 0 and this determinant is also zero so your rank being three or less than three depends upon this Delta correct now try to observe one thing if you take this Matrix a if data is equal to zero then this all third order minus will also be zero because the rank of this augmented Matrix depends only and only on this determinant because all this they are zeros by default now tell me one thing can I say one thing here we don't get any case like rank of your not equal to rank of a augmented B correct why because this is a rank of a augmented B depends upon this determinant which is again depending on that a so if a data is 0 then a rank of a will be something less than here also this will be something Less Than 3 if data is not 0 then a bank of a is 3 this is also not zero a augmented B is also three so here we don't have any case like rank of a is not equal to a rank of a augmented B now if you see this condition if you see this condition this case is ruled out s or no means oh I think something happened okay fine so if you see one thing carefully in this case of homogeneous system of equations this third case is not anymore valid or getting valid means these two will be always same but it depends depending upon the debt value it has unique or infinitely many but this case cannot happen in case of a homogeneous systems what is the meaning of that the system always has solution S1 so homogeneous systems always has a solution and the solution is nothing but x equal to 0 y equal to 0 and Z equal to zero if you replace all these questions to be zeros yeah so if you see this equation if you replace all this x y z values with zeros then definitely this equations get satisfied so in case of homogeneous systems it's never a case like the system don't have a solution the system always has a solution and the solution is x equal to zero y equal to zero and Z is equal to zero Okay so here you can see rank of a will be always equal to rank of a augmented B that's why which means you cannot always rank of a is equal to a rank of a augmented B which means the system always has a solution the homogeneous system you can write better the homogeneous system the homogeneous system has ah you know always has a the homogeneous system always has a solution always has a solution okay so now if you see this what is that solution X is equal to zero zero zero and this solution is called AVL solution a real solution what is the meaning of table independent it is solution Always by default for when you put x equal to 0 y equal to 0 and Z is equal to zero you need not bother about what are the nine coefficient values correct so this solution always acts as a solution for the given homogeneous system of matrices okay and this solution is called tavial solution now let us suppose if data is not equal to zero for example if data is not equal to 0 then what happens a rank of a will be equal to rank of a augmented B and that is equal to number of unknowns then the system will have unique solution obviously so that unique solution is nothing but this solution now if the homogeneous system for the homogeneous system for the homogeneous system to have a non-trivial solution to have a non AVL solution to have a non-trivial solution means apart from 0 comma 0 if it want to have some other solution then you know definitely rank of a less than n n is basically number of equal unknowns if you see this implies determinant of a should be definitely zero correct because a is a NBN Matrix if a is NBA and Matrix and if rank of a has to be less than n then definitely this determinant has to be 0 and this point is very very important point for get examination that for the homogeneous system to have a non trivial solution the determinant of the coefficient Matrix should be zero why this is important because they give you question like this they'll give you three homogeneous equations and out of these nine coefficients they do not give one coefficient value and they'll ask you for a given system to have a non-davel solution what is the code constant or the the unknown value so basically what you will do you equate the determinant to zero so you'll have one equation with one unknown and you will you know write the solution of this clear so this is how we decide whether the system has you know whether the system has a solution or you don't have a solution generally clear so this is basically the talk about system of equations almost half now we have discussed on system of equations so normally if you take non homogeneous systems depending upon the ranks of a and a augmented B you decide how many solutions does the set of equations have actually clear so this is one very important part of this linear algebra and the main Crux of this linear algebra is talking about this vectors or you can say uh this equation is basically here clear I hope you understood this so with this we close this path which is system of equations and uh you know basically this ranks of the matrices okay now I would like to talk okay before going for one last part which is eigen values and eigenvectors I would like to talk something on linear dependent vectors and linearly independent uh vectors basically okay so let us talk about that firewall okay now see carefully here I'll talk about linearly dependent and independent vectors linearly dependent and linearly dependent and independent vectors linearly dependent and independent vectors let us see what are this linear dependent and independent vectors why I am calling vectors because normally if you take 3D space for example if you are talking in three dimensional space if you want to draw a position Vector for example let us suppose X why and Z let us take in this direction a right-handed coordinate system so let us suppose if you want to take a point in space let us suppose this my finger tip is a point in space which I want to talk about then if I connect this origin to this fingertip through a vector then definitely this point has some coordinates in space A B C correct so for three coordinates x coordinate y coordinate and Z coordinate now these three coordinates can be represented as a matrix x y z that's why we deal generally vectors in matrices okay so now if you see anyhow if X1 X2 X3 and so on x n r n row or column vectors what is column vectors basically these are column matrices if if a matrix has only one column it is called a column Matrix if a matrix has only one O It's called aromatics basically okay so now if x 1 x 2 x 3 and so on X N column vectors vectors and and K1 K2 K3 and so on k n scallus n scalars then the combination and then the combination then the combination K1 X1 K1 X1 plus k 2 x 2 plus KT x 3 and so on k n x n x n is called linear combination of this vectors linear combination of vectors these are called linear combination of vectors okay now you can understand one thing basically what is linear just we are changing the size we are not changing any direction so if you see this Vector is Multiplied with this scalar K1 so that the magnitude of this Vector changes but the direction does not change correct let us suppose if I take this finger to this Vector let us suppose if I two times of this Vector the length changes but the magnitude I mean the direction of the vector doesn't change okay so that's what I am telling you this is called linear combination of vectors now this combination is said to be linearly dependent or independent depending on a particular equation I will show you now okay if if K 1 x 1 plus k 2 x 2 plus K 3 x 3 and so on k n x n is equal to 0 for K 1 k 2 K3 and so on k n not all zeros simultaneously not all zeros simultaneously then we say combination is linearly dependent we say combination combination is combination is linearly dependent combination is linearly dependent look let me explain you the help of example you will understand for example okay so now let us suppose I have taken three vectors example let us suppose I can take three vectors x 1 is equal to say one two three for example x 2 is equal to some Something 2 1 4 something like this anything X they for example let us take X Y is 1 minus 1 1 for example something like this okay now try to understand one thing let us suppose if I write linear combination some constant into this plus some constant into this plus some constant into this then K 1 x 1 plus k 2 x 2 plus K 3 x 3 this will become linear combination now let us suppose K1 into X1 plus K2 into X2 plus K3 into this is equal to zero let us suppose I'm adding this equation so these are some constraints which are to be multiplied and these are three by one matrices now when all these are not zeros when all these things are not zeros if this equation is getting satisfied then we say that the that these vectors are depending on each other how come look let us suppose if I keep one if I keep this value minus 1 and if I keep this value minus 1 okay for example if I keep this value or plus one let us suppose I am keeping Plus 1. plus 1 then when K 1 k 2 K 3 these are all not zeros but still this equation is holding good s or No 1 minus 2 minus 1 minus one plus one zero means basically here I mean this is the Matrix actually the null Matrix zero zero zero okay I have written capital O that's why if you see so this is zeros similarly if you take 2 minus 1 1 minus 1 0 3 minus 4 minus one plus one zero so here if you see this combination is getting satisfied when K 1 k 2 K they are non zeros actually clear so such combinations are called linearly dependent you know this word dependent very often let us suppose if I give you some equation Y is equal to some function in terms of X then you say that Y is depending on x correct so that's basically dependency and here the vectors are linearly multiplied so that's why this combination is called linear combination and here if you see I can write this equation as look can I write this equation as 1 into x 1 x 1 minus 1 into X2 minus x 2 plus x 3 is equal to 0 so which means for example if I write expression for X2 I can write this as X2 is equal to x 1 plus X3 similarly if I want to write x 3 I'll write this as x 2 minus X1 similarly I can write expression for X1 also as you know X2 minus x 3 like this so here if you see you can write this vectors as depending on some other vectors okay this can happen only if these constants are non-zeros look let us suppose if all these constants are zeros K 1 k 2 K all are zeros then definitely the equation will get satisfied but you cannot write any relation like this you cannot show any dependency of vectors on the other vectors so that's why this combination is called linear combination with linear dependent vectors now I will ask you one thing you can understand easily look when I write these vectors if I take this vectors x 1 x 2 x they determinant let us suppose if I take the determinant x 1 x 2 x 3. means basically my intention is to tell you that let X1 is equal to a 1 b 1 c 1 x 2 is equal to A2 B2 C2 x 3 is equal to B3 and C3 let us suppose these are the vectors if X1 X2 X3 are dependent are linearly dependent you can say are linearly dependent dependent now tell me one thing first determinant of this x 1 x 2 x 3 means basically x one is one column Vector a 1 b 1 c 1 this is another column Vector a two B two c two this determinant should be definitely equal to 0 why it should be equal to 0 because you know that this X2 is nothing but some combination of X1 and X3 similarly if you see if for example if I write this transformation column 1 changes to column one minus column two plus column three for example if I write this combination C1 changes to C1 minus C2 plus C3 then what you will get here x 1 minus x 2 plus x 3 which is actually 0 in this case Okay so the First Column if you apply this transformation to this determinant your First Column will be 0 so definitely this determinant is zero so whenever these vectors I have taken the T by J K simply similarly same thing holds Good by four by four five by five all these things actually okay so if you see carefully if certain vectors are linearly dependent then you have to do make sure that this determinant is always zero and why this determinant is 0 because you can see this whenever things are depending linearly we get such equations and if you apply the corresponding transformation means if you apply this C1 changes to C1 minus C2 plus C3 this First Column gets out to be 0 so that's called linear combination of vectors and this case is said to be linearly dependent okay and obviously if we cannot write any such dependency we'll call the cases linearly independent okay so linearly independent if you write linearly independent if you write this a combination the linear combination the linear combination K 1 x 1 plus k 2 x 2 and so on plus k n x n is said to be please said to me linearly independent linearly independent if if K 1 x 1 plus k 2 x 2 plus 3 and so on plus k n x n is equal to 0 when K1 equal to K2 is equal to KT and so on k n is equal to zero so whenever all these constants are 0 so you cannot write any linear dependency of a vector on other vectors for example you cannot say that x 3 is equal to x 1 minus x 2 anything like that okay and why I have told this about linear independency my I want to give one many point here number of linearly independent Solutions number of linearly independent Solutions number of linearly independent Solutions of ax is equal to B is number of unknowns minus number of unknowns unknowns minus a rank of a basically here okay so this point is very important basically what happens when you get infinitely many solutions then few Solutions will be linearly depending on each other and few will not be depending on each other okay so if you want to find out what are the number of linearly independent Solutions you have a simple formula for that number of unknowns minus a rank of a I just want to give you this formulation so it's necessary for me to make you understand what is linearly dependent and what is linearly independent so I have talked about this linear dependency and independency a bit for example okay so now we'll go for this last module of this or the last part of this video which is eigen values and eigenvectors which is again one very most questions asked region in linear algebra okay so we have discussed about this linear dependent and linear independent combinations of vectors now let us go to one next and the last most important part which is eigen values and eigenvectors eigen values and eigen vectors eigen values and eigen vectors so let us see what are these eigen values and eigen vectors these are practically very important vectors actually now what are these vectors actually look let us try to understand so normally you see the definition that of a minus Lambda is equal to zero all these things that's characteristic polynomial but why this debt of a minus Lambda I should be equal to 0 this is what you need to understand means how the characteristic equation is coming out okay so for writing that you should first of all know what what is the definition of an eigen Vector okay so let us write eigen vector eigen vector so the definition of this eigen Vector is basically any column vector any column vector X basically this X is not equal to 0 means basically it's not a zero column Vector so any column Vector that satisfies the equation that satisfies the equation that satisfies the equation a x is equal to Lambda X for some scalar Lambda for some scalar Lambda for some scalar Lambda and basically this Lambda is called eigen value of this x Lambda is called eigen value of a eigen value of a and X is called X is called the corresponding eigen Vector the corresponding the corresponding the corresponding eigenvector of Lambda eigen Vector of Lambda okay so this is eigenvector of Lambda now basically if you see this Matrix X is basically one matrix it's a column Matrix now what's happening if you take any Matrix and if you transpose on this means if you are multiplying these two matrices means if you are applying this a transformation on this x you see the outcome is again x correct so if you write this in Vector form like this for example you will understand it better whenever you are applying any transformation on a vector the outcome is in the same direction but magnitude is changing because depending on the Lambda Lambda is scalar so scalar multiplication of a vector will always keep the direction same correct so if you see these X vectors are such special vectors such that when you apply transformation of this a on this Vector then the direction of the vector remains same but only the magnitude changes now try to understand one thing and we have this equation ax is equal to Lambda x correct so if you simplify this equation a minus Lambda times of I into X is equal to 0 here correct so a minus Lambda times of I into X is equal to zero now tell me one thing this equation or this is basically system of homogeneous equations correct so this is system of homogeneous system of homogeneous equation okay so this is System of homogeneous equation now try to understand one thing if this system has to have a non-trivial solution of course when x is 0 then this equation automatically gets what you have seen in previous case when ax is equal to 0 whenever this Vector is basically 0 Vector then this equations gets satisfied automatically that's called a trivial solution but we have clearly defined that this x is non-zero if this x is non-zero that means what do I mean if this x is non-zero this implies for eigen vectors to exist for eigen vectors to exist for eigenvectors to exist this system should have non-trivial solution so no and you know five homogeneous system to have a non-table solution determinant of the coefficient Matrix which means this year in this case so debt of Lambda this determinant of this Matrix should be definitely equal to 0 and this is how you get the characteristic equation and this equation is called basically characteristic equation characteristic equation characteristic equation and the roots of the characteristic equation the roots of the characteristic equation so basically if you see this is a polynomial in terms of Lambda so if you solve you will get the values of Lambda so the roots of the characteristic characteristic equation is called roots of the characteristic equation are called eigenvalues are called are called eigen values basically okay so these are called eigen values so basically how we will calculate the eigen values of any Matrix you know you will equate this determinant to zero but why that determinant is 0 because this system of equations should have a non-trivial solution understood so that's the reason why this determinant should be equal to zero so obviously once I give you the square Matrix then what you can do you can write a minus Lambda times of I you can get the determinant U equal to 0 then you will get some polynomial in terms of lambdas and once you solve that polynomial you will be getting the values for this uh you know eigen values okay so basically this is how we calculate the eigen values and how basically we will discuss some properties some important properties of eigen values here okay so properties of eigen values properties of eigen values properties of eigen values properties of eigen values look the first property that I will give you here is the eigen values of the eigen values of lower dangle lower triangular comma upper triangular upper triangle comma diagonal matrix diagonal matrices are the principal diagonal elements of the Matrix are the principle diagonal principle diagonal elements diagonal elements of The Matrix principle diagonal elements of the Matrix okay so this is how we have the result because obviously it's way you might know this very well because if a is upper triangular or lower triangular Matrix then a minus Lambda I will also be upper triangular or lower triangle because this Lambda I subtraction will only make changes to the principle diagonal elements correct so if the Matrix is by default upper triangular or lower triangular a minus Lambda I will also be upper and lower triangles so the update is nothing but product of the principle diagonal elements basically means if I want to explain you for example if I take a diagonal matrix a minus Lambda I will be like this 5 2 by 2 I'm talking about so 0 0 8 3 3 minus Lambda is equal to zero okay example I'm talking about one example here so if you see if you want to calculate the determinant of this Matrix again the product so this implies a 1 1 minus Lambda into a two two minus Lambda into a minus Lambda this is equal to Z over here okay so if that is equal to 0 then obviously you know this is already in factorized form so if this has to be 0 Lambda can be a 1 1 or a22 or a33 so basically these are the things which we have uh you know uh principal diagonal elements of the given Matrix okay so this is one very very important property and this property you know uh generally many times you will be using that in examinations okay and second I would like to give you second property the sum of eigen values of a matrix the sum of eigen values of M at X the sum of eigen values of a matrix is equal to is equal to trace of The Matrix the Rays of Matrix so basically what is Trace of a matrix this is equal to sum of principal diagonal elements sum of principle diagonal elements sum of principle diagonal elements basically here okay so this is equal to sum of the principal diagonal elements means basically if you want to mention this Sigma I is equal to 1 to n let us suppose if you have n by n Matrix then you'll have n eigen values so this is equal to Sigma I is equal to 1 to n a i i that's it okay so a one one plus a 2 2 plus a33 and so on a and N okay now coming to the third very important property the product of eigen values the product of eigen values of a matrix of a matrix is equal to again it's very important and very simple property of course is equal to the determinant of The Matrix is equal to the determinant of a matrix is equal to the determinant of a matrix okay means basically if you take the product if you take the product notation product I is equal to 1 to n Lambda I if you take this this is nothing but determinant of a actually here okay so these two properties are very much important and many times you can expect a question from this that sum of the eigen values is actually equal to the sum of the principal diagonal elements and product of the eigen values is equal to the you know determinant of that particular Matrix okay now I'd like to give you some more uh good information the eigen values of the eigen values of a and a transpose a and a transpose are same because you see this means basically if you take this eigen values of a and a transpose if you see the diagonal elements remain the same and of course the determinant basically determinant of a minus Lambda or a transpose minus Lambda I that remains same okay so that's how the characteristic polynomial is same and eigen values will also be same okay now 5 if Lambda 1 Lambda 2 Lambda 3 and so on Lambda n are eigen values of a eigen values of a eigen values of a then first point you have certain points here so first point eigen values of eigen values of a power m are Lambda 1 power m Lambda 2 power m Lambda 3 power M and so on Lambda n power and so on this result will basically go on and so on Lambda n power M okay so when you raise the power of a matrix then automatically the eigen values also raise their power and second so when M value is equal to minus one one special case of a inverse when eigen values of a inverse are 1 by Lambda 1 because Lambda 1 whole power minus 1 gives you 1 by Lambda 1 1 by Lambda 2 1 by Lambda 3 and so on this goes on to 1 by Lambda n okay so here you can observe one thing let us suppose if Matrix is singular means let us suppose if the determinant doesn't exist if mat X is singular that means definitely at least one eigen value would have been definitely zero cut because you see determinant is nothing but product of the eigen values so if determinant is zero definitely one of the eigen values should be definitely 0 so that's how here if you are writing a inverse that means you are very much sure that no eigen value is zero so they all are in the denominators okay so no issues next third eigen values of eigen values of some constant K 1 into a plus some uh B some K2 into I let us say let us suppose if you have some combination K 1 into a and K2 into I are K1 times of Lambda 1 plus K2 comma K1 times of Lambda 2 plus K2 comma and so on K 1 Lambda n plus K2 this is how we have the eigen values of some Advanced you know terms basically here okay and a and a transpose has the same eigen values of course okay next fourth one eigen values of a joint a eigen values of a joint a r debt a by Lambda 1 this property is very important okay sometimes they ask you this Lambda 2 Delta by Lambda 3 and so on that a by Lambda n okay so this property is again uh you know very much important property generally many times they try to frame a question on this so eigen values of this adjoint AR of this form the T by Lambda 1 that a by Lambda 2 and so on basically they exist like that okay and so generally these are the things which could be available for you and now I would like to go for the sixth property which is again some good property 0 is definitely an eigenvalue 0 is definitely an eigenvalue of an odd orders Q symmetric Matrix an odd order skew symmetric Matrix basically okay so 0 is definitely an eigen value of orders Q semantic Matrix means basically if you see you know that every Square symmetic Matrix of odd order has zero determinant so if that is zero definitely one of the eigen values should be zero okay so this is one very good statement because now we know Logics of all these things it's not like you by heart something or you just take it for granted you know why 0 is definitely an eigen value because for every odd order skill symmetic Matrix your determinant should be zero clear now let us continue seven eigen values of skew symmetric Matrix or you can write eigen values of real symmetric Matrix symmetic matrices if you take symmetric Matrix normally matters are always a l always real okay next eight property I could give you something like eigen values of eigen values of you know eigen values of skew symmetric matrices if you talk about Q symmetric eigen values of geosymmetric Matrix are either 0 either there will be zeros in case of odd order or purely imaginary what's the meaning of purely imaginary if you take any complex number you know basically uh complex numbers have of this form Z is equal to X Plus i y if you take this as the complex number then you know X is called the real part Y is called the imaginary part so here in these cases this x value will be 0 okay you'll have only the Y part so your complex numbers come out in the form like I into some constant basically okay so that's how you have this eigen values of excuse me take Matrix one important property I give you here 9 eigen values of eigen values of eigen values of ah you say eigen values of for example we can talk about orthogonal Matrix eigen values of orthogonal Matrix Matrix are of unit modulus of unit modulus means basically what's the meaning of that if they have real numbers there will be either plus or minus 1 if they are complex numbers then they turn out to be some complex numbers of magnitude one okay means basically if you calculate root of x square plus y square that x square plus y Square will be 1 actually here okay now the minimum properties actually but at the gate level uh these are properties which they generally ask you but coming to some Advanced levels linear algebra is a fantastic subject and I would like to give and thank actually one person right now because you know by this time you might have passed already three years or the end of us of video but uh all this knowledge what I've got is from one person Gilbert strong he has a very good book on linear algebra linear algebra by Gilbert twang he is basically from MIT and you know he is fantastic purpose across the uh he's known for his Works in linear algebra across the world okay so you can also find him on some YouTube or somewhere in MIT openware courses anyway so he's really a fantastic guy okay so eigen values of orthogonal Matrix of unit modulus basically and yeah if you keep on there are certain many other things like for example every eigenvalue lies within the cycle of certain areas of where the radius is defined by some sum of two other elements okay we have certain things like that very importantly but right now coming to the gate uh you know examination level all these points could be definitely helpful for you these four five points what are given here then these points and finally these two points are definitely a lot many questions in eigenvalues comes from these two and of course the first one is a direct thing that you can talk of okay so yes so generally these are the things that that they come off so tenth uh okay so fine I think we had a lot of enough eigen values and how to calculate this eigen vectors okay so how to calculate this eigenvectors I'll show you shortly uh because we will deal with one very important concept which is called diagonalization of matrices okay because many of you might be writing engineering Sciences paper as well so this diagonalization is one very important Concept in matrices okay we'll see what is this diagonalization okay look so eigenvector so if you write properties of eigen vectors properties of eigenvectors properties of eigen vectors okay so properties of eigenvectors so if you see we will write one thing eigen vectors of of a and a power M both will be same are same because what is the difference between them look I'll tell I'll make you understand one thing normally eigenvector means ax is equal to Lambda x correct now let us suppose if you make this power M then what happens let us suppose if you have a power M then the eigenvectors corresponding to this a power M will be just Lambda power M times of this x yes or no look this Lambda poem is just a scalar and it don't actually uh you know change the things so that's why this a power M if you apply transpose on this this will again match each other so that's how you actually have the eigenvectors of a and a power M both same second I would like to give you this so second is eigenvectors of a and a transpose are I would like to change the color because this is important not same okay uh not same actually here and one very good property I would like to give off symmetric matrices eigen vectors of symmetric matrices foreign vectors of symmetric matrices are always orthogonal are always mutually orthogonal are always mutually orthogonal okay so this matrices are basically mutually orthogonal actually okay and means basically let us suppose for three mathematics you'll find any given eigenvalue you can have any number of eigen vectors but if you take any two eigenvectors corresponding to two different values of Lambda then definitely this dot product is going to be 0 okay so now let us try to understand this this eigenvectors of symmetric matrices are always mutually orthogonal now I would like to give one very important Point here number of linear independent eigenvectors number of linearly independent eigenvectors of a Matrix a for example is equal to is equal to number of distinct eigen values of a number of distinct eigen values of a means basically for example let us suppose if in a table Matrix you got only two different eigen values one eigenvalue is repeating then the number of eigen independent eigen vectors is basically two in that case Okay so number of distinct eigen values of a is what you can have here generally okay so basically these are the you know basic common elements of the syllabus into all the branches in general they talk with the linear algebra then come to the bank of the Matrix determinants eigenvalue eigenvectors again one very important topic okay so basically it's majority portion is covered in this video fire some quite good time and I hope this video will definitely help you out uh in answering your gate questions and uh basically to develop some knowledge on this linear algebra as well okay so I wish you all a very good luck and thank you for being patient for almost some four hours of time in this lecture and definitely this is going to be a very good you know boost up for all your linear algebra and this is a single short video so definitely after this you know uh I felt I believe that you definitely would have gained some confidence in this video and in this linear algebra topic and uh very soon we'll be coming up with the next topic which is basic calculus thank you all and I wish you all a very good luck for your preparation thank you