CHAPTER 05.19: SYSTEM OF EQUATIONS: Inverse of matrices Example

In this segment we will look at how we can show that one matrix is the inverse of the other matrix. We’re not showing how to find the inverse of a matrix, but just to see that can you show that - hey - one matrix is inverse of another one. So let’s suppose the problem statement says: hey - determine if a [B] matrix, for example a two by two matrix 3, 2, 5, 3, is inverse of this particular matrix [A] matrix, which is written as -3, 2, 5, -3. So what one of the things which I can do is I can take in order to show that - hey whether [B] is the inverse of [A] is I can take [B] multiply by [A] and see if it’s turning out to be the identity matrix. So I put 3, 2, 5, 3 here and then I’ll put -3, 2 here 5, -3 here and of course I have two by two matrix being multiplied by another two by two matrix. So I’ll get a two by two matrix.

So what I can do is I can in order to find out this element right here. I’ll take this row and multiply it by this column right here so first row multiplied by the first column will be the first row first column of the identity matrix so it’s 3 times -3 is -9, 2 times 5 is plus 10 so I get a 1 here. First row multiplied by the second column right here will be the first row second column right here 3 times 2 is 6, 2 times – 3 is – 6. So that gives me 0. So similarly I can show this to be 0 and this to be 1 and that shows that hey [B] is in fact inverse of [A] because that itself is the identity matrix of two rows and two columns. I could’ve also done the problem by doing [A] times [B] I didn’t have to do [B] times [A] I could’ve done [A] times [B]. So in that case I have -3, 2, 5, -3 multiplied by 3, 2, 5, 3 so that’s my [A] and that’s my [B] and I could again if you follow the multiplication of two matrices.

For example this one this element right here will be first row first column right here so -3 times 3 is plus 9, 2 times 5 is 10 so that addition gives me 1 similarly I get 0 here, 0 here, and a 1 here so that itself is the identity matrix, which is a two by two matrix. So that’s how you if someone tells you that hey do you mind if one matrix is the inverse of the other one; the first thing you which have to be clear about is that it’s a square matrix and then you either multiply the one matrix by the another one or the other matrix by the other one and both of them should turn out to be identity matrix. But you have to show only one of them; this is an extra - this is another way of proving the same thing so please don’t think that this also has to be shown so you can either show [B] times [A] is the identity matrix or [A] times [B] is the identity matrix. That is enough to show that the two matrices are inverse of each other. And that’s the end of this segment.