CHAPTER 05.21: SYSTEM OF EQUATIONS: Finding the inverse of a matrix Theory

In this segment we’ll talk about how we will find the inverse for a matrix. If A is a n by n matrix, so if we have a square matrix and we claim that A inverse is the inverse of A, then we know that hey A times A inverse will be equal to the identity matrix. So the question arises that how do we find out the inverse of a matrix so at least get an idea of how we should go about doing that. So if we consider the A matrix, let’s suppose for a n by n matrix to be of this form, so we have A one one going all the way to A one n. A n one going all the way to A and n so we have these n by n elements, n squared elements in the A matrix. And then if we have the A inverse matrix and that also has n square elements let’s call it A one one prime all the way up to A one n prime. Then A n one prime and A nn prime.

Then how will we be able to find what the inverse of a matrix is? All we have to do is to multiply this matrix by this matrix and put it into the identity matrix. So that’s how we’ll be able to do that. So if we have this matrix here we have A one one, all the way up to A one n then A n one all the way up to A nn I am going to multiply it by the inverse of a matrix which will be A one one prime, A one n prime, A n one prime all the way up to A nn prime. And I’m going to put that equal to the identity matrix. So I get one, zero, all the way to zero, zero, one, zero and all the way to zero here and one here. So what I basically have is that our n by n matrix which I’m multiplying by its inverse and I’m supposed to get the identity matrix right here. So the question arises: how do I find out what the inverse of the matrix. What I can look at at this phase is that now I can now rewrite finding the inverse of the matrix as a system of simultaneous linear equations as follows. I can say hey let me take the first column of the inverse matrix. So if I take the first column of the inverse matrix and I multiply it to the A matrix, it’ll basically give me the first column of the identity matrix. Which makes sense because if I take the first row here and first column here, I’ll get the first row first column here. If I take the first row here, second row here and first column here, I’ll get the second row first column here. If I take the nth row here and the first column here I’ll get the nth row first column here. So what that’s going to do is that if I now write down only the first column here of the inverse matrix then ill only get a column matrix on the right side also. So this is A one one and it goes to the A one n, and this is A n one and that’s A nn I’ll write this as A one prime all the way up to A n one prime. So that’s my first column of the inverse of the matrix. Then this one will just be one zero, zero, zero. So what that means is in order to find the first column of the inverse of the matrix, I’ll solve n equations and unknowns which will be of this particular matrix form.

And what does that mean in terms of finding the other columns of the inverse of the matrix? If I would have put the second column of the inverse of the matrix here, then in this case the right hand side would become zero one zero zero zero so on and so forth. And so what that means is that I’ll have the second column of the identity matrix would be my right hand side vector. So if I continue doing that I’ll be able to set up an equation n unknowns but our n such sets of n equations n unknowns and I’ll be able to find the inverse of the matrix. And that is the end of this segment.