Hello students, once again welcome you to GK's Applied Maths classes. Today what we are going to do is we are going to use Excel and do matrix and determinant operations like addition, subtraction, multiplication. transpose of a matrix finding the determinant value and finding inverse.
Okay. You can also use this video as a reference to create your record book, which is a practical for class CBSE class 12 applied maths. Okay. Okay.
And for non-applied math students, those who are taking pure maths also, this will be more of an enrichment. Okay. So I'll be using Excel now. So I'll be sitting and doing and explaining. I hope you will enjoy this video.
Okay. Straight away let's go to Excel. Okay. I'm in Excel. It will take few minutes.
Just a second. Okay, let me start. Now what we are going to do is we are going to add two matrix.
So I will take the matrix A as, let me increase the font size. So I will take matrix A as, I will take a 2 by 3 matrix. So just enter 2, 3, minus 1. and then it is 3, 4 and 1. So I have, I will just increase the size a bit, 24. It is a 2 by 2 matrix.
Followed by I will take B, my matrix B will be, what is my matrix B? I will take it as 1, minus 1, 2. and 3 1 & 2 so we have created two matrix okay which is matrix a and matrix B both are of same order since they are of same order it should be possible for us to add now how do I find a plus B so I am finding a plus B okay now how do I do that Now A plus B, the order should be 2 by 3. So first what you do, select the 2 by 3 column. See first I select. You see look here.
I go here and I select a 2 by 3 matrix. order of a and order of b is 2 by 3 so i select my a plus b of order 2 by 3 then i i type equal to equal to stands for the formula equal to since i want to do addition all that i need to do is select this entire matrix a select the matrix a plus Select the second matrix B. Select the matrix A followed by the second matrix B.
You see this is basically A plus B. Don't press enter. You hold control shift enter.
Now you can see you have got A plus B. Just have a look at it. It is A plus B. Okay, very clear now, it's A plus B. I will just do A minus B also for you.
A minus, one second, A minus B. Now how do I do A minus B? Let me select this as 24 so that you can see clearly.
A minus B, what you do? Select the matrix A. First of all, you have to select a 2 by 3 matrix.
First you select a 2 by 3 matrix. For a formula, it is always equal to Select the matrix A. See I select the matrix A. That is C4 to E5.
Since I want to subtract put a minus sign. And again you select the matrix B. You can't press enter. It has to be control shift and enter.
Now you can see it is A minus B is very clear. Now I also want to find, I want to find, just a second, I want to find 2A plus 3B. I want to find 2A plus 3B.
How do I do that? For the same matrix A and B. So first what you do is select the 2 by 3 matrix first you select the 2 by 3 matrix equal to sign Since it is 2A, you put 2 star. Star stands for into. You can put it in a bracket.
It's better. Select the matrix A. Close the matrix.
You want to add. Put a plus sign. Again 3 star. Open the bracket. Select the matrix B.
Close the bracket. Control shift enter. You get, clearly you get 2a plus 3b. So let's try 2a minus 3b.
Let's say 2a minus 3b. 2a or you can say 3a minus 2b. Now how do I go about it? First of all it should be a 2 by 3 matrix equal to sign. 3 into it's better to put a bracket select 3a that is a close bracket subtraction is minus sign 2 star it's B close bracket control shift enter something's wrong let me check okay I must have made some mistake bracket bracket problem so control shift enter perfectly so we have learned how to add and subtract not only add and subtract we also learned how to do scalar multi if you want I'll just show you only to you see I want to find 2 a okay now how do I find 2 a the 2a will be like it's a 2 by 3 matrix equal to 2 star put a bracket select the matrix control shift enter now you can see clearly 2a matrix 24. So, see 2, you can see, this is multiplied by 2. 4, 6, minus 2, 6, 8 and 2. I hope this is clear.
This is how you do addition of 2 matrix, subtraction of 2 matrix and scalar multiplication of 2 matrix. Now I am going to teach you how to multiply 2 matrix. Now how do I multiply 2 matrix? Suppose if I have a matrix A, my matrix A, let me take it as 2 minus 1, oh sorry, 2. Minus 1, say 3 and this is 3, 2 and 4. This is matrix A and I am taking matrix B. What is my matrix B?
My matrix B I will take it as 2 minus 4. 1, minus 1, 3 and 2. It is a 3 by 2 matrix. So let me increase the font size so that you can see clearly. 24 should be ok. Now I want to find A.
How do I find A? I want to find matrix A, sorry matrix A. Now clearly A is of order. 2 by 3. B is of order 3 by 2. So my resultant matrix will be of order 2 by 2. So first you select a 2 by 2 matrix.
First you select a 2 by 2. If you don't select, you won't get the answer. You have to select a 2 by 2 matrix. Equal to, okay.
Now there is a command. M mult. That means matrix multiplication.
Can you see that? M mult. So click on that. Okay. So first you select the first matrix.
Comma. Very important. Then select the second matrix.
Close the bracket. Control, shift, enter. So I got the multiplication.
Very beautiful. Control shift enter. You got the matrix.
You can see clearly. You see 2 into 2 is 4. 4 minus 1 is 3. And 9. 9 plus 3 is 12. So it's perfect. It works.
Let us try to find what is B A's. B A. Now clearly your BA matrix will be of what order? It will definitely be of order 3 by 3. Why it is 3 by 3? Because B is of order 3 by 2, A is of order 2 by 3, so the resultant matrix should be of order 3 by 3. So first I select the 3 by 3 matrix, see I have selected it.
Then usual equal to matrix multiplication, easy to remember, M mult means matrix multiplication, select it. Now you have to select B matrix first, comma. A matrix. Close the bracket.
Don't forget it. Control shift and enter. Now you can see you have got the multiplication done.
So this is how you have learned how to add, how to subtract, how to multiply two matrix. Okay. I will just do one more question with the 3 by 3 matrix multiplication.
A equal to the matrix is, this is one of the board question. 2 minus 3 minus 5. minus 1, 4, 5, 1, minus 3 and minus 4, followed by the matrix B. What is my matrix B?
It's 2, minus 2, minus 4, minus 1, 3, 4. 1, minus 2 and minus 3. So let me just make it big so that you can see clearly. So these are the two matrix. Now I want to find A, B. How do you find A, B? Clearly both are 3 by 3 so the resultant matrix will be of order 3 by 3. So first you select a 3 by 3 matrix and what is the command? to young malt selected select the first matrix comma select the second matrix close the bracket control shift enter Interestingly, this is nothing but matrix A.
So this is one of the famous problem that we have. 24. Let us see it is exactly matrix A. Let us see what is B A is.
How do I find B A? Select a 3 by 3 matrix equal to matrix mult. select the B matrix first comma select the a matrix close the bracket control shift enter you see it is nothing but B matrix I will make it a little big for you so they can see clearly this is a very famous question that which came in the board exam also so you have learned how to multiply two matrix I gave you multiplication of two by three 3 by 2 now I am giving you multiplication of as 2 square matrix okay now finally what we are going to do I am also going to talk to you about how to find the determinant value how to find determinant suppose i want to find the determinant value of matrix a i want to find determinant value of matrix a so how do i do matrix a take a matrix a i want to find determinant value so equal to equal to stands for formula m d e t the moment you put m d e t it gives you m determine that means determinant select it select the determine A close the bracket control shift enter so the determinant value is 0 okay here the determinant value is 0 I'll increase the size for you 24 determinant value is 0 you can cross check you by usual way I want to find determinant value of B now how do i find determinant value of b equal to m determinant m determinant select select this close control shift enter interestingly both the values are zero okay let me make it 24 right this is determinant of p and this is determinant of a okay now let us also try to find what is the determinant value of a b is okay equal to but that is the same determinant it has to be zero right equal to m determinant m determinant select this Control shift enter it has to be 0. So you are proved.
Determinant of a b is same as determinant a into determinant b. So you are able to prove that result. Determinant of a b is determinant a into determinant of b. Determinant a b is 0. determinant a is 0 determinant B is also 0 what do you think will be determinant of B that also will be 0 we'll find out equal to M determinant select select the determinant BA close the bracket control shift enters it will also be 0 isn't it isn't it amazing Let me just say this is, that's fine.
So these are the different determinant values. Now let's go to the next part. How to find inverse of a matrix? So how to find inverse of a matrix?
So let's go to inverse of a matrix. Let me enter the matrix A equal to, sorry, A equal to, what are the matrix? 3, 4, 2, 0, 2, minus 3, 1, 1, minus 2 and 6. So, first of all I want to know what is the determinant values, okay.
So, let us check whether inverse exist or not. so let's check that this is determinant of a that's what i'm going to find determinant of a that is equal to m determinant the same formula m determinant select this matrix close the bracket control shift enter it is 2 definitely determinant value is not 0 so inverse will exist so let us find out what is a inverses in of a now how do I find inverse of a inverse is also a 3 by 3 matrix first to select the 3 by 3 matrix then you put equal to as usual the formula is equal to since you want to find multiple matrix inverse it is yeah I end you see automatically to give you M inverse can you see it M inverse select that now select the array select the 3 by 3 matrix close the bracket control shift enter you've got the inverse okay so i found the inverse so this is how you find the inverse of a matrix okay now let me take inverse of this inverse of this matrix inverse of inverse of a just to verify okay just so first of all select a 3 by 3 matrix so that you get used to that formula select a 3 by 3 matrix equal to m inverse m inverse so m inverse select this matrix what answer are you expecting definitely it should be a right let us see control shift enter so you got the matrix here can you see it of course because of approximation the sum error will do this it's two point something okay you can always format the cell and make it to number if i put number it will give me with no decimal basis there are a lot of other things you can do with it so now the value is zero can you see it there will be some because of uh approximation there will be some error so you have i hope you understood how to find inverse of a matrix okay now finally i'm going to talk to you about transpose suppose i have a matrix a uh and i will take a two by three three by two matrix uh three minus one 3 followed by 2 minus 4 6 okay and i have a matrix b equal to it should be minus 1 6 minus 7, I am sorry it should be 2 by 3, it should be minus 7 down, minus 7, 4, then what we will do, 2 and minus 3, okay. Now what I will do first is, I want to find transpose of a matrix. If A is a 2 by 3 matrix, we will just select it. If A is a 2 by 3 matrix, so what will be A transpose?
Definitely its order will be 3 by 2, correct? If it is a 2 by 3, the order will be 3 by 2. So first you select a 3 by 2 matrix. very important equal to young P are sorry you just type transpose immediately get transpose no need to put M transpose it transpose transpose of what transpose of a select the matrix close control shift enter so automatically get the transpose now I want to find B transpose how do I find B transpose It should be a 2 by 3 matrix because my B is 3 by 2, so it has to be 2 by 3. So equal to transpose.
Select it. Select the matrix B. Close the bracket.
Control. shift enter wonderful isn't it so you got b transpose now i want to find a b a b how do i find a b this is a two by three matrix this is a three by two matrix so a b will be a two by 2 matrix, it will be a 2 by 2 matrix equal to M mult that means matrix multiplication I am sorry I was mistake equal to M mult. Select it. Select the first matrix A, comma. Select the second matrix B, close the bracket, control, shift, enter.
So I got my A. Now I want to find A whole transpose. Now what is A whole transpose? It's a 2 by 2 matrix, so it is transpose. Select A, close the bracket, control shift enter.
So I got my A transpose. Now what I want to do is I want to find B transpose into A transpose. B transpose into A transpose. So how do I do that? First you select a 2 by 2 equal to select my B transpose.
i'm sorry it should be multiplication so it should be m mult m mult select b transpose comma a transpose close the bracket control shift enter so you have clearly proved one of the most important in matrix that A transpose is same as B transpose into a transpose okay I hope with that we will stop I think I have performed all operations I have taught you how to do to add two matrix, okay. We have learnt how to add two matrix, subtract two matrix, then I learnt, we taught you how to find 2a plus b, 2 3a minus 2b, then I found out how to find scalar of a matrix, then we learnt how to multiply a matrix, a b and b a, okay, and then we learnt how to find determinant value and after that we went to find... when to find inverse of a matrix and how to find determinant value and finally transpose i hope this video is useful to you kindly subscribe to my channel and also click the bell icon to get notifications of all video this will definitely be helpful for you for doing your practicals in your class 12 applied maths okay see you again the next video have a nice day bye