Introduction to Matrices and Their Operations

Good morning dear students. So today we are going to start a new session of matrices. Actually I have done the same chapter in other language. Okay, so I am just doing this series in English language now for few people who cannot understand Hindi or the other languages. Okay, so matrices. Let us without wasting a minute, let us directly go into the concept. So first thing that comes into your mind is what is matrix, right? This is the plural, this is singular. So what is a matrix? That is nothing but a rectangular arrangement. I will not tell you the definitions because you are going to get them in the textbook as well. So I'll just tell you a normal way, rectangular arrangement. So what you are going to do, you're taking certain, they can be numbers, they can be alphabet, no problem. So usually we take numbers, they can be real numbers, they can be complex numbers as well. So if you are taking real numbers that becomes a real matrix. If you are taking complex numbers then that becomes a complex matrix. So here I am just taking a real matrix because at the basic level you do not consider complex numbers. When you go to the higher level you consider complex matrices as well. So see these things this is a rectangular arrangement. Why rectangle? Because it see the way we have arranged these numbers it appears as a rectangle. See it is like rectangle right. So that is the reason rectangular arrangement of certain numbers they can be real or complex is nothing but a matrix. So in matrix there comes one more concept that is row and one more column. These two concepts most of the students even don't know this thing means they feel confusion or they are not aware of this thing then you can even get it here see row means are this way. In the sense the things the entries which are written in this way entries in the sense whatever I have taken inside these brackets okay these things are known as elements or entries of a matrix okay of any particular matrix we call these are the entries of a matrix or elements of a matrix. So rows in the sense which are written in horizontal way column in the sense which are written in the vertical way. So here 1, 2, 3 which is written in the horizontal way this way okay so that is that comprises first row then 4, 5, 6 this comprises second row okay and I said column or vertical right so this one 1, 4, 1, 4 this is the first column okay I have written in explanation form sorry expanded form if you want you can write like c1 it is just for your understanding okay. C2 means which one? Second column. C3 is 3, 6. Okay, so this is first column, second column, third column. So this is the concept of row and column. So next we shall move on to one more concept that is known as order. Order of a matrix, right? So order of a matrix. So usually we represent that as m cross n. or m by n m by n m cross n okay not an issue so how you are going to write this thing is any matrix suppose i take the same matrix 1 2 3 4 5 6 then what is the order of this matrix if i say matter let me just represent it as matrix a okay so what is the order of this matrix then you have to say number of rows these are number of rows whatever dm this is number of columns okay so first you have to write number of rows that is m means 1 2 2 rows right r 1 r 2 when it comes to column c 1 c 2 c 3 c 3 right so number of rows are 2 cross or by number of columns 1 2 3 so the order of this particular matrix which is given to us is 2 by 3 or 2 cross 3. Okay. So, this is one more basic concept that you must be knowing. Then let's just move on to types of matrices. See, there is I have just represented these three numbers. Now, if we take particular entries, then what the particular matrix is noun has. Okay. It is known as with some unique name. So, let us just go with types of matrices to understand the various types of matrices. So, types of matrices. The first one that we are going to learn is a row matrix. What do you mean by row matrix any matrix where there is only a single row number of columns. can be more than one but the row is always one so how do you represent say one two three four five you can write any numbers of any number of columns okay but the row is this one r1 only what about columns c1 c2 c3 c4 c5 there are five columns okay so what is the order of this matrix it is number what is the order i said number of rows cross number of columns so number of rows 1 number of columns 5 right because number of rows is always number of rows is always 1 in row matrix that is the reason the general order we write for this one is 1 cross n n can be any number okay that depends on which example you are taking this is row matrix similarly we have one more thing that is column matrix you can easily guess from the above definition that here number of Rows can be any okay any number of rows can be written but number of columns is always 1. Okay so how we are going to write is 1 2 3. Suppose I write like this number of rows are 1 or 2 or 3 this is c1. Okay so what how many number of rows 3 cross 1. So this is the matrix of order 3 cross 1. Here this one is constant always. So that is why we have written this order. Okay. So then comes one more matrix, zero matrix. So by the name itself, you can guess what I actually mean. See zero, this is matrix. You're going to write matrix in which all the entries are zero. All right. So this is also known as null matrix in mathematics or anywhere null in the sense nothing. Right. So zero matrix. It is represented as O is equal to you write any number of entries you can write I have just taken 4 it is 2 cross 2 matrix ok. So, 4 zeros if I take some more 0 is equal to 0 0 0 0 I can take like this as well ok this is 1 2 3 cross 2. So, 0 matrix in the sense the matrix in which all the elements are 0 that is known as 0 matrix then. comes the next type of matrix which is the most important that is a square matrix okay so we're just having one more matrix it is square matrix okay so what is a square matrix it is a matrix in which number of rows is equal to number of columns if i write the order of any matrix just like this or like this that value of m and n must be equal then that becomes a square matrix let me just explain you with an example so m is equal to n i just write two rows okay and two columns 1 2 1 2 okay see r 1 r 2 c 1 c 2 so what's the order of this matrix is 2 cross 2 okay so the matrix in which number of rows is equal to number of columns is known as square matrix okay the elements can be you know the entries which you are taking can be anything it's your wish you can take any number okay then that becomes a square matrix but in a square matrix okay if you take particular entries then that is known as particular with particular name let us just go with those also but before that let me tell you one more concept so in a square matrix okay you Here if I take in 1, 2, 3, 3 columns then I have to take 3 rows. Okay. So in a square matrix these particular elements which are in this way. Okay. These elements they are known as. Okay. Di. I will just write it here for your convenience. Diagonal elements. They are known as diagonal elements. And this particular thing. okay this is called principal diagonal also they can be called as principal diagonal elements as well if you are taking other elements that will be off diagonal elements okay so why i said the concept of diagonal elements is because you can understand the next type of matrix by using this okay so the next type of matrix which is also a square matrix but with particular condition let us just try to know that as well so that is diagonal matrix So what is a diagonal matrix? It's a square matrix. Okay, it's a square matrix. But what is the condition in this square matrix is that the Elements other than diagonal elements, okay. So, you can say like diagonal elements are non-zero, okay, but the elements other than the diagonal are all, okay, let me just write it. Elements other than diagonal elements, diagonal elements, okay, are zero. So, what do you mean by this? See, if I write a square matrix, 1, 2, 3, 4, 5, 6, 7, 8, 9, then these diagonal elements are non-zero. This condition is satisfied, but all the other elements, see, elements other than, elements or you can take entries as well, no problem, entries other than diagonal elements are zero, right? So, this is diagonal, these are diagonal elements. So, other than that, these are zero, zero, zero, zero, zero, zero, zero. So, this is a 3 cross 3 square matrix. So here this is this becomes a diagonal matrix when the diagonal elements are non-zero and all the other elements are zero. If I write 2 cross 2 matrix then diagonal elements are non-zero other elements are zero. Simple okay. If I write 4 cross 4 matrix just have a notice 1 0 0 0 2 0 0 0 0 3 0 0 0 0 4. So the diagonal elements here are non-zero. ok they are non-zero and all the other elements are all the other entries are zero so this is 4 cross 4 ok so this is known as diagonal matrix square matrix in which these condition must must satisfy ok now diagonal matrix in that you take one more consider you consider one more condition that is these diagonal elements are all same if i take 2 it is 2 2 2 2 ok so then that particular matrix is known as one more name that is scalar matrix. So, what is a scalar matrix? What is a scalar matrix? It is actually a diagonal matrix. Okay. What do you mean by diagonal matrix? You have learnt it. So, all the other elements other than diagonal elements must be 0. But here the condition is that the diagonal elements a i i if I take is equal to one particular number. They should be always equal to one particular number in the sense they should all be always equal. Suppose I take the same example 4 x 4 matrix. If I write 2 0 0 0 2 0 0 0 0 0 3 0 0 0 0 2. This is not a scalar matrix as because the diagonal elements are not same. 3 is there which is a different value. So if I take this as 2 then this becomes a scalar matrix. I hope you have got this point. Scalar matrix in the sense the matrix the diagonal matrix in which all the diagonal elements are same. Okay. So, in that you do not consider k is equal to k is not equal to 1. k should not be equal to 1 because why because you will come to know in next definition that if diagonal elements are equal to 1. Okay. Scalar matrix in which diagonal elements are equal to 1 is known as unit matrix okay there is one more name unit matrix or identity matrix identity matrix so that is the reason you do not take k is equal even if you take that is not a mistake okay but because we already have one standard name for this matrix where k is equal to 1 we do not include this here okay you can take any number of values so this is unit matrix next you have one more matrix let us just go with that as well so that is known as upper triangular matrix so it means that it is also a square matrix actually where what happens is that suppose this is one particular matrix square matrix right M is equal to n so M cross M okay so here the diagonal elements and the up above elements They form upper triangle in the sense all the entries here okay are non-zero but the elements below are all zero. So then that is known as upper triangular matrix. What actually happens here is that let me just tell you an example 1 2 3 0 4 5 0 0 6. See the elements diagonal and the other elements they are triangle. All the elements below the diagonal are 0 then that is known as a per triangular matrix. Now you can guess the other matrix definition as well lower triangular matrix. matrix it is just has upper triangular opposite to it okay so what here here in the matrix the elements are in the lower triangular here there are non-zero elements here the elements are zero okay so let me take the same example one 0 0 2 3 0 4 4 5 6 okay so see this is forming a triangle in the sense the elements below the diagonal elements are non-zero and above are 0 that is known as lower triangular and if I take upper triangular and lower triangular combined then that is known as triangular matrix which is nothing but again diagonal matrix right upper triangular and lower triangular when combined together forms a diagonal matrix itself right because upper triangular means the elements below this must be 0 diagonal must be 0 lower triangular means the elements above the diagonal elements must be 0 and when you you take triangular it is both upper and lower so that means the elements below the diagonal below the diagonal and above the diagonal both are zero then therefore the only diagonal elements are left that is nothing but our standard definition diagonal matrix okay so next we have Let us go with one more definition that is we have sub matrix. S U B sub matrix. I hope you are all familiar with subset right. So if I take a big set A B C D something like this and if I take some part. only b c then this particular thing is the subset of this cm applies to the matrices as well so sub matrix is nothing but a matrix that is obtained from a standard matrix suppose i have here a matrix 1 2 3 4 5 6 7 8 9 okay so if i get one matrix from this one by eliminating one row and one column okay it is not standard definition that you should eliminate one column only you can eliminate any number of columns and any number of rows but when you eliminate you obtain one certain the other matrix right suppose here i am obtaining 5 6 8 9 so this becomes sub matrix of this matrix E okay so b is the sub matrix of a then you have one more concept that is known as principal sub matrix when do you call it as principal sub matrix what is the difference you must be knowing this thing as well sub matrix in the sense you can eliminate any number of rows and any number of columns to obtain a matrix which may be square which may not be a square matrix okay here coincidentally i have obtained a square matrix let me take one more example of four cross four matrix okay. So, I have this matrix. What is the order of this matrix? Is that 1, 2, 3, 1, 2, 3, 4, okay, 3 cross 4. So, here this is matrix A. Then if I eliminate 1, and one column I am going to get matrix which is 7 7 7 0 3 4 okay so what is the order of this 1 2 1 2 3 so this is the sub matrix of this this matrix but if I eliminate the number of rows and number of columns in such a way that the sub matrix that I obtain is a square matrix. Let me just tell you illustrate this with an example you eliminate one row and you eliminate two columns then the sub matrix that you are getting is 7734 okay 7734. So this is a principal sub matrix I hope you have got the difference between the sub matrix and the sub matrix. means you can obtain a mat you know a matrix by eliminating any number of rows and any number of columns that matrix may be square matrix may not be the square matrix okay may not be the square sub matrix okay so but the sub matrix which you obtain by eliminating number of rows and number of columns is a square matrix you have obtained square sub matrix then that particular is known as principal sub matrix okay i hope you have got this thing then let us go with one more concept that is equality of matrices suppose i take two matrices a is equal to a b and 1 2 okay i'll just check in general example b is equal to i am taking c t 1 2 okay and now to call these two matrices a is equal to b if i am calling this a is equal to b that simply implies that a A must be equal to C strictly implies that A must be equal to C, B must be equal to D. d it simply means that if you want to call any matrix equal to any other matrix then the corresponding elements must be equal what do you mean by corresponding elements corresponding element in the sense if you are taking first element of a matrix then and comparing with the other matrix then you must compare to the first element of the other matrix if you are taking second element you must compare it with the second element itself you cannot compare b is equal to fourth element okay so that is why If you have to call any two matrices as equal, the corresponding elements must be equal. Okay. So this is just simple concept. Next, let us come to addition of the matrices. So addition, I hope you know, you know, the normal addition, right? Addition of the simple thing applies here as well. Okay. So addition, if I tell you two plus three, what do you say? It's fine, right? The same way, if I'm telling A plus B, two matrices. If I take 1, 2, 3, 4 is matrix A plus 1, 2, 5, 7 is the other matrix then add these two matrices. Okay if I curl this you have to add these two matrices then what you are going to take is that you are again going to add corresponding element that is the first element with the first element 2, 2 plus 2, 4, 5, 3, 8, 4, 7. 4 7 is 11 right here you can observe one more condition that in order to add any two matrices they must be of same order always remember this condition this condition is known as confirmability of addition of matrices okay whether two matrices are confirmed for addition or not it depends on the order of those two matrices okay you So these two are 2 cross 2 2 cross 2 and the resultant obtained is also 2 cross 2. Okay. So then this if they are of same order then they get they are confirmed to add or else you cannot add. Why? Let me take this example. See 1 2 3 4 5 6. What's the order of this? Two rows, three columns and I take this matrix 1 2 5 7. Okay. So this is 2 cross 2. Now corresponding elements if you are going to add this column. cannot be add cannot be added why because there is no third column right so that is the reason two matrices when we are adding them when we are adding the two matrices the order of those two must be same okay so then a similar way if we go with multiplication multiplication of matrix by scalar multiplication by a Scalar what do you mean by scalar is nothing but it is a constant number it can be 2 it can be 3 it can be 4 but it's a number okay that is not a scalar so you're going to multiply matrix suppose I say 2a then the whole if 1 2 3 4 is matrix a then multiply 2 to every element okay then 2 1s are 2 2 2s are 4 2 3s are 6 2 4s are 8 this is matrix multiplication by scalar Then let me tell you the same concept I mean a concept in the same matrix itself that is trace trace of a matrix that is always trace of a square matrix. So, what does this mean is that adding the diagonal elements that's it. Suppose trace of this matrix if I tell you 8 only diagonal elements 8 plus 2 is equal to so we will denote it as tr of a is equal to 10. tr of a is equal to 10 okay so trace of a is equal to 10 if i take some other matrix suppose i take this matrix then 1 plus 7 is equal to 8 so tr of that particular matrix is b then it is 8 okay so that is a nice trace of matrix which is nothing but adding the diagonal elements so i hope you are done with the okay there is there are some more concepts that is transpose of the matrix and other concepts i think the video will be too long oh Okay, let me just try to continue this thing. Let me explain it a bit faster. If I have a matrix A, 1, 2, 3, 4, then the transpose of this matrix is nothing but write the rows as columns. Okay, so 1, 2, you write it as 1, 2, 3, 4, write it as column. So, this is known as A transpose. So, transpose is nothing but you are going to change the rows and columns, interchanging of the rows and columns. That process is known as transposition. Okay. But when you transpose that, that particular is A transpose. It is denoted as A power T or you can write simply A dash. problem okay. So when you find any particular matrix where the matrix and its transpose are equal then that matrix has a particular name that is known as symmetric matrix. So let us just take an example 1 6 7 6 5 2 7 2 3. So this is a matrix you just do the transpose of this matrix by yourself and you will obtain that that is again equal to the same matrix. okay so that is known as symmetric matrix next let us come to the skew symmetric matrix very symmetric matrix very simple diagonal elements can be anything but these are equal okay mirror images okay then when you come to skew symmetric matrix you are going to write diagonal elements must be zero this is the condition and all the other numbers are mirror especially mirror elements okay there you are taking mirror elements they are equal but with opposite sign if i take 3 here it must be minus 3 if i take plus 6 it should be minus 6 so when you do the transpose of this matrix it is equal to minus of particular given matrix okay so when this condition is satisfied this is known as q symmetric matrix okay i hope you have got this point uh if you have not got i'll explain it once again okay just drop a comment i'll explain this it's very simple actually uh fine Then there's this let's end up with one concept last concept that is product of matrices. How do you multiply two matrices is also a very important concept. See when you're taking any matrix suppose I take 1 2 3 4 and have one more matrix 1 2 3 4 5 6. See for addition I said that the order must be same. Then for multiplication what is the condition is that number of see here 2 cross 2. and here 2 cross 2 cross 3 right so 2 cross 2 here number of columns number of columns of the first matrix must be equal to number of rows you just write the order if this is equal you write you know m cross n here it should be n cross p then the answer that you obtained that matrix will have the order m cross p always remember this condition you So how you're going to multiply is an important thing. Let me just explain it. You have to multiply this first row with the first column. Simple. The same first row with the second column, the same first row with the third column. Okay, so how do you do that? 1 into 1 plus 2 into 4, right? Then you are going to get the first element of the resultant matrix which will have m cross p order in the sense 2 cross 3. Okay, so next first row with the second column that is 1 into 2 plus 2 into 5. Similarly, 1 into 3 plus 2 into 6. Next, second row elements with the first column, second row elements with the second column, second row elements with the third column. just as you did the first row so we'll write it as i've just written them directly to save the time okay so this is how you're going to write and then what you're going to do is that you just write the total answer 1 into 1 is 1 plus 4 to the 8 is 9 similarly you just solve by yourself you'll get this answer right 19 26 and 24 okay 33 so this is having the order 2 cross 3. So, most important thing for matrix multiplication. So, any two matrices can be multiplied only when the number of columns of the first matrix is equal to number of rows of the second matrix. This is the important condition which these matrices must satisfy. Then matrix multiplication is possible. Okay. So, I think this is enough for today. We will just sum up everything now. Okay. So, we have learned about matrices we will learn about types of matrices right so what is the matrix rectangular representation then you're having rows you have the concept of column just please if you have any doubt please drop a comment i'll just try to answer those things okay next uh let's just wrap up we'll come with some other video soon so this is enough for today thank you so much god bless you

Good morning dear students. So today we are going to start a new session of matrices. Actually I have done the same chapter in other language.

Okay, so I am just doing this series in English language now for few people who cannot understand Hindi or the other languages. Okay, so matrices. Let us without wasting a minute, let us directly go into the concept.

So first thing that comes into your mind is what is matrix, right? This is the plural, this is singular. So what is a matrix? That is nothing but a rectangular arrangement.

I will not tell you the definitions because you are going to get them in the textbook as well. So I'll just tell you a normal way, rectangular arrangement. So what you are going to do, you're taking certain, they can be numbers, they can be alphabet, no problem.

So usually we take numbers, they can be real numbers, they can be complex numbers as well. So if you are taking real numbers that becomes a real matrix. If you are taking complex numbers then that becomes a complex matrix.

So here I am just taking a real matrix because at the basic level you do not consider complex numbers. When you go to the higher level you consider complex matrices as well. So see these things this is a rectangular arrangement.

Why rectangle? Because it see the way we have arranged these numbers it appears as a rectangle. See it is like rectangle right.

So that is the reason rectangular arrangement of certain numbers they can be real or complex is nothing but a matrix. So in matrix there comes one more concept that is row and one more column. These two concepts most of the students even don't know this thing means they feel confusion or they are not aware of this thing then you can even get it here see row means are this way.

In the sense the things the entries which are written in this way entries in the sense whatever I have taken inside these brackets okay these things are known as elements or entries of a matrix okay of any particular matrix we call these are the entries of a matrix or elements of a matrix. So rows in the sense which are written in horizontal way column in the sense which are written in the vertical way. So here 1, 2, 3 which is written in the horizontal way this way okay so that is that comprises first row then 4, 5, 6 this comprises second row okay and I said column or vertical right so this one 1, 4, 1, 4 this is the first column okay I have written in explanation form sorry expanded form if you want you can write like c1 it is just for your understanding okay. C2 means which one? Second column.

C3 is 3, 6. Okay, so this is first column, second column, third column. So this is the concept of row and column. So next we shall move on to one more concept that is known as order.

Order of a matrix, right? So order of a matrix. So usually we represent that as m cross n. or m by n m by n m cross n okay not an issue so how you are going to write this thing is any matrix suppose i take the same matrix 1 2 3 4 5 6 then what is the order of this matrix if i say matter let me just represent it as matrix a okay so what is the order of this matrix then you have to say number of rows these are number of rows whatever dm this is number of columns okay so first you have to write number of rows that is m means 1 2 2 rows right r 1 r 2 when it comes to column c 1 c 2 c 3 c 3 right so number of rows are 2 cross or by number of columns 1 2 3 so the order of this particular matrix which is given to us is 2 by 3 or 2 cross 3. Okay.

So, this is one more basic concept that you must be knowing. Then let's just move on to types of matrices. See, there is I have just represented these three numbers.

Now, if we take particular entries, then what the particular matrix is noun has. Okay. It is known as with some unique name. So, let us just go with types of matrices to understand the various types of matrices. So, types of matrices.

The first one that we are going to learn is a row matrix. What do you mean by row matrix any matrix where there is only a single row number of columns. can be more than one but the row is always one so how do you represent say one two three four five you can write any numbers of any number of columns okay but the row is this one r1 only what about columns c1 c2 c3 c4 c5 there are five columns okay so what is the order of this matrix it is number what is the order i said number of rows cross number of columns so number of rows 1 number of columns 5 right because number of rows is always number of rows is always 1 in row matrix that is the reason the general order we write for this one is 1 cross n n can be any number okay that depends on which example you are taking this is row matrix similarly we have one more thing that is column matrix you can easily guess from the above definition that here number of Rows can be any okay any number of rows can be written but number of columns is always 1. Okay so how we are going to write is 1 2 3. Suppose I write like this number of rows are 1 or 2 or 3 this is c1.

Okay so what how many number of rows 3 cross 1. So this is the matrix of order 3 cross 1. Here this one is constant always. So that is why we have written this order. Okay. So then comes one more matrix, zero matrix. So by the name itself, you can guess what I actually mean.

See zero, this is matrix. You're going to write matrix in which all the entries are zero. All right.

So this is also known as null matrix in mathematics or anywhere null in the sense nothing. Right. So zero matrix.

It is represented as O is equal to you write any number of entries you can write I have just taken 4 it is 2 cross 2 matrix ok. So, 4 zeros if I take some more 0 is equal to 0 0 0 0 I can take like this as well ok this is 1 2 3 cross 2. So, 0 matrix in the sense the matrix in which all the elements are 0 that is known as 0 matrix then. comes the next type of matrix which is the most important that is a square matrix okay so we're just having one more matrix it is square matrix okay so what is a square matrix it is a matrix in which number of rows is equal to number of columns if i write the order of any matrix just like this or like this that value of m and n must be equal then that becomes a square matrix let me just explain you with an example so m is equal to n i just write two rows okay and two columns 1 2 1 2 okay see r 1 r 2 c 1 c 2 so what's the order of this matrix is 2 cross 2 okay so the matrix in which number of rows is equal to number of columns is known as square matrix okay the elements can be you know the entries which you are taking can be anything it's your wish you can take any number okay then that becomes a square matrix but in a square matrix okay if you take particular entries then that is known as particular with particular name let us just go with those also but before that let me tell you one more concept so in a square matrix okay you Here if I take in 1, 2, 3, 3 columns then I have to take 3 rows.

Okay. So in a square matrix these particular elements which are in this way. Okay. These elements they are known as.

Okay. Di. I will just write it here for your convenience.

Diagonal elements. They are known as diagonal elements. And this particular thing.

okay this is called principal diagonal also they can be called as principal diagonal elements as well if you are taking other elements that will be off diagonal elements okay so why i said the concept of diagonal elements is because you can understand the next type of matrix by using this okay so the next type of matrix which is also a square matrix but with particular condition let us just try to know that as well so that is diagonal matrix So what is a diagonal matrix? It's a square matrix. Okay, it's a square matrix.

But what is the condition in this square matrix is that the Elements other than diagonal elements, okay. So, you can say like diagonal elements are non-zero, okay, but the elements other than the diagonal are all, okay, let me just write it. Elements other than diagonal elements, diagonal elements, okay, are zero. So, what do you mean by this?

See, if I write a square matrix, 1, 2, 3, 4, 5, 6, 7, 8, 9, then these diagonal elements are non-zero. This condition is satisfied, but all the other elements, see, elements other than, elements or you can take entries as well, no problem, entries other than diagonal elements are zero, right? So, this is diagonal, these are diagonal elements. So, other than that, these are zero, zero, zero, zero, zero, zero, zero. So, this is a 3 cross 3 square matrix.

So here this is this becomes a diagonal matrix when the diagonal elements are non-zero and all the other elements are zero. If I write 2 cross 2 matrix then diagonal elements are non-zero other elements are zero. Simple okay.

If I write 4 cross 4 matrix just have a notice 1 0 0 0 2 0 0 0 0 3 0 0 0 0 4. So the diagonal elements here are non-zero. ok they are non-zero and all the other elements are all the other entries are zero so this is 4 cross 4 ok so this is known as diagonal matrix square matrix in which these condition must must satisfy ok now diagonal matrix in that you take one more consider you consider one more condition that is these diagonal elements are all same if i take 2 it is 2 2 2 2 ok so then that particular matrix is known as one more name that is scalar matrix. So, what is a scalar matrix?

What is a scalar matrix? It is actually a diagonal matrix. Okay. What do you mean by diagonal matrix? You have learnt it.

So, all the other elements other than diagonal elements must be 0. But here the condition is that the diagonal elements a i i if I take is equal to one particular number. They should be always equal to one particular number in the sense they should all be always equal. Suppose I take the same example 4 x 4 matrix. If I write 2 0 0 0 2 0 0 0 0 0 3 0 0 0 0 2. This is not a scalar matrix as because the diagonal elements are not same.

3 is there which is a different value. So if I take this as 2 then this becomes a scalar matrix. I hope you have got this point. Scalar matrix in the sense the matrix the diagonal matrix in which all the diagonal elements are same.

Okay. So, in that you do not consider k is equal to k is not equal to 1. k should not be equal to 1 because why because you will come to know in next definition that if diagonal elements are equal to 1. Okay. Scalar matrix in which diagonal elements are equal to 1 is known as unit matrix okay there is one more name unit matrix or identity matrix identity matrix so that is the reason you do not take k is equal even if you take that is not a mistake okay but because we already have one standard name for this matrix where k is equal to 1 we do not include this here okay you can take any number of values so this is unit matrix next you have one more matrix let us just go with that as well so that is known as upper triangular matrix so it means that it is also a square matrix actually where what happens is that suppose this is one particular matrix square matrix right M is equal to n so M cross M okay so here the diagonal elements and the up above elements They form upper triangle in the sense all the entries here okay are non-zero but the elements below are all zero. So then that is known as upper triangular matrix. What actually happens here is that let me just tell you an example 1 2 3 0 4 5 0 0 6. See the elements diagonal and the other elements they are triangle.

All the elements below the diagonal are 0 then that is known as a per triangular matrix. Now you can guess the other matrix definition as well lower triangular matrix. matrix it is just has upper triangular opposite to it okay so what here here in the matrix the elements are in the lower triangular here there are non-zero elements here the elements are zero okay so let me take the same example one 0 0 2 3 0 4 4 5 6 okay so see this is forming a triangle in the sense the elements below the diagonal elements are non-zero and above are 0 that is known as lower triangular and if I take upper triangular and lower triangular combined then that is known as triangular matrix which is nothing but again diagonal matrix right upper triangular and lower triangular when combined together forms a diagonal matrix itself right because upper triangular means the elements below this must be 0 diagonal must be 0 lower triangular means the elements above the diagonal elements must be 0 and when you you take triangular it is both upper and lower so that means the elements below the diagonal below the diagonal and above the diagonal both are zero then therefore the only diagonal elements are left that is nothing but our standard definition diagonal matrix okay so next we have Let us go with one more definition that is we have sub matrix. S U B sub matrix. I hope you are all familiar with subset right.

So if I take a big set A B C D something like this and if I take some part. only b c then this particular thing is the subset of this cm applies to the matrices as well so sub matrix is nothing but a matrix that is obtained from a standard matrix suppose i have here a matrix 1 2 3 4 5 6 7 8 9 okay so if i get one matrix from this one by eliminating one row and one column okay it is not standard definition that you should eliminate one column only you can eliminate any number of columns and any number of rows but when you eliminate you obtain one certain the other matrix right suppose here i am obtaining 5 6 8 9 so this becomes sub matrix of this matrix E okay so b is the sub matrix of a then you have one more concept that is known as principal sub matrix when do you call it as principal sub matrix what is the difference you must be knowing this thing as well sub matrix in the sense you can eliminate any number of rows and any number of columns to obtain a matrix which may be square which may not be a square matrix okay here coincidentally i have obtained a square matrix let me take one more example of four cross four matrix okay. So, I have this matrix. What is the order of this matrix?

Is that 1, 2, 3, 1, 2, 3, 4, okay, 3 cross 4. So, here this is matrix A. Then if I eliminate 1, and one column I am going to get matrix which is 7 7 7 0 3 4 okay so what is the order of this 1 2 1 2 3 so this is the sub matrix of this this matrix but if I eliminate the number of rows and number of columns in such a way that the sub matrix that I obtain is a square matrix. Let me just tell you illustrate this with an example you eliminate one row and you eliminate two columns then the sub matrix that you are getting is 7734 okay 7734. So this is a principal sub matrix I hope you have got the difference between the sub matrix and the sub matrix. means you can obtain a mat you know a matrix by eliminating any number of rows and any number of columns that matrix may be square matrix may not be the square matrix okay may not be the square sub matrix okay so but the sub matrix which you obtain by eliminating number of rows and number of columns is a square matrix you have obtained square sub matrix then that particular is known as principal sub matrix okay i hope you have got this thing then let us go with one more concept that is equality of matrices suppose i take two matrices a is equal to a b and 1 2 okay i'll just check in general example b is equal to i am taking c t 1 2 okay and now to call these two matrices a is equal to b if i am calling this a is equal to b that simply implies that a A must be equal to C strictly implies that A must be equal to C, B must be equal to D.

d it simply means that if you want to call any matrix equal to any other matrix then the corresponding elements must be equal what do you mean by corresponding elements corresponding element in the sense if you are taking first element of a matrix then and comparing with the other matrix then you must compare to the first element of the other matrix if you are taking second element you must compare it with the second element itself you cannot compare b is equal to fourth element okay so that is why If you have to call any two matrices as equal, the corresponding elements must be equal. Okay. So this is just simple concept. Next, let us come to addition of the matrices. So addition, I hope you know, you know, the normal addition, right?

Addition of the simple thing applies here as well. Okay. So addition, if I tell you two plus three, what do you say?

It's fine, right? The same way, if I'm telling A plus B, two matrices. If I take 1, 2, 3, 4 is matrix A plus 1, 2, 5, 7 is the other matrix then add these two matrices.

Okay if I curl this you have to add these two matrices then what you are going to take is that you are again going to add corresponding element that is the first element with the first element 2, 2 plus 2, 4, 5, 3, 8, 4, 7. 4 7 is 11 right here you can observe one more condition that in order to add any two matrices they must be of same order always remember this condition this condition is known as confirmability of addition of matrices okay whether two matrices are confirmed for addition or not it depends on the order of those two matrices okay you So these two are 2 cross 2 2 cross 2 and the resultant obtained is also 2 cross 2. Okay. So then this if they are of same order then they get they are confirmed to add or else you cannot add. Why? Let me take this example.

See 1 2 3 4 5 6. What's the order of this? Two rows, three columns and I take this matrix 1 2 5 7. Okay. So this is 2 cross 2. Now corresponding elements if you are going to add this column. cannot be add cannot be added why because there is no third column right so that is the reason two matrices when we are adding them when we are adding the two matrices the order of those two must be same okay so then a similar way if we go with multiplication multiplication of matrix by scalar multiplication by a Scalar what do you mean by scalar is nothing but it is a constant number it can be 2 it can be 3 it can be 4 but it's a number okay that is not a scalar so you're going to multiply matrix suppose I say 2a then the whole if 1 2 3 4 is matrix a then multiply 2 to every element okay then 2 1s are 2 2 2s are 4 2 3s are 6 2 4s are 8 this is matrix multiplication by scalar Then let me tell you the same concept I mean a concept in the same matrix itself that is trace trace of a matrix that is always trace of a square matrix.

So, what does this mean is that adding the diagonal elements that's it. Suppose trace of this matrix if I tell you 8 only diagonal elements 8 plus 2 is equal to so we will denote it as tr of a is equal to 10. tr of a is equal to 10 okay so trace of a is equal to 10 if i take some other matrix suppose i take this matrix then 1 plus 7 is equal to 8 so tr of that particular matrix is b then it is 8 okay so that is a nice trace of matrix which is nothing but adding the diagonal elements so i hope you are done with the okay there is there are some more concepts that is transpose of the matrix and other concepts i think the video will be too long oh Okay, let me just try to continue this thing. Let me explain it a bit faster.

If I have a matrix A, 1, 2, 3, 4, then the transpose of this matrix is nothing but write the rows as columns. Okay, so 1, 2, you write it as 1, 2, 3, 4, write it as column. So, this is known as A transpose.

So, transpose is nothing but you are going to change the rows and columns, interchanging of the rows and columns. That process is known as transposition. Okay. But when you transpose that, that particular is A transpose. It is denoted as A power T or you can write simply A dash.

problem okay. So when you find any particular matrix where the matrix and its transpose are equal then that matrix has a particular name that is known as symmetric matrix. So let us just take an example 1 6 7 6 5 2 7 2 3. So this is a matrix you just do the transpose of this matrix by yourself and you will obtain that that is again equal to the same matrix. okay so that is known as symmetric matrix next let us come to the skew symmetric matrix very symmetric matrix very simple diagonal elements can be anything but these are equal okay mirror images okay then when you come to skew symmetric matrix you are going to write diagonal elements must be zero this is the condition and all the other numbers are mirror especially mirror elements okay there you are taking mirror elements they are equal but with opposite sign if i take 3 here it must be minus 3 if i take plus 6 it should be minus 6 so when you do the transpose of this matrix it is equal to minus of particular given matrix okay so when this condition is satisfied this is known as q symmetric matrix okay i hope you have got this point uh if you have not got i'll explain it once again okay just drop a comment i'll explain this it's very simple actually uh fine Then there's this let's end up with one concept last concept that is product of matrices. How do you multiply two matrices is also a very important concept.

See when you're taking any matrix suppose I take 1 2 3 4 and have one more matrix 1 2 3 4 5 6. See for addition I said that the order must be same. Then for multiplication what is the condition is that number of see here 2 cross 2. and here 2 cross 2 cross 3 right so 2 cross 2 here number of columns number of columns of the first matrix must be equal to number of rows you just write the order if this is equal you write you know m cross n here it should be n cross p then the answer that you obtained that matrix will have the order m cross p always remember this condition you So how you're going to multiply is an important thing. Let me just explain it.

You have to multiply this first row with the first column. Simple. The same first row with the second column, the same first row with the third column. Okay, so how do you do that? 1 into 1 plus 2 into 4, right?

Then you are going to get the first element of the resultant matrix which will have m cross p order in the sense 2 cross 3. Okay, so next first row with the second column that is 1 into 2 plus 2 into 5. Similarly, 1 into 3 plus 2 into 6. Next, second row elements with the first column, second row elements with the second column, second row elements with the third column. just as you did the first row so we'll write it as i've just written them directly to save the time okay so this is how you're going to write and then what you're going to do is that you just write the total answer 1 into 1 is 1 plus 4 to the 8 is 9 similarly you just solve by yourself you'll get this answer right 19 26 and 24 okay 33 so this is having the order 2 cross 3. So, most important thing for matrix multiplication. So, any two matrices can be multiplied only when the number of columns of the first matrix is equal to number of rows of the second matrix.

This is the important condition which these matrices must satisfy. Then matrix multiplication is possible. Okay.

So, I think this is enough for today. We will just sum up everything now. Okay.

So, we have learned about matrices we will learn about types of matrices right so what is the matrix rectangular representation then you're having rows you have the concept of column just please if you have any doubt please drop a comment i'll just try to answer those things okay next uh let's just wrap up we'll come with some other video soon so this is enough for today thank you so much god bless you

Transcript for:Introduction to Matrices and Their Operations

Transcript for:
Introduction to Matrices and Their Operations