CHAPTER 05.21: SYSTEM OF EQUATIONS: Finding the inverse of a matrix Example   In this segment we will look at how to find the inverse of matrix through an example. Let’s suppose somebody says hey go head and find out the inverse of this matrix 25, 5, 1, 64, 8, 1, 144, 12, 1 so I will see from the definition of the inverse of a matrix if I multiply this square matrix by its inverse I will get the identity matrix of course we are assuming here that the inverse exists. So we’re not looking into that if we face any problems in finding the solution to the set of equations then we know that hey we have some problems with finding the inverse of the matrix, but let’s suppose in this case it is already given to you that the inverse exists so this will be a a 1 1 prime, a 1 2 prime, a 1 3 prime, a 2 1 prime, a 2 2 prime, a 2 3 prime, so on and so forth. So this is the matrix that you should find for an inverse this is the inverse of the matrix so this is [A] this is [A] inverse and this will be the identity matrix 1, 0, 0, 0, 1, 0, 0, 0, 1.   So if we’re going to solve for the inverse of the matrix what we can do is we can only look at the first column let’s suppose of the inverse matrix, then the answer will be the first column here. If we want to find out the second column of the inverse of matrix then your answer will be the second column of the identity matrix; if I want to find the third column of the inverse of a matrix I will use the third column of the identity matrix as my right hand side by able to doing that I will be able to set up my equation. So let’s go ahead and see how that turns out to be the case. So what we have is we have 25, 5, 1, 64, 8, 1, 144, 12, 1 and then the first column of the inverse matrix is a 1 1 prime, a 2 1 prime, and a 3 1 prime and that’ll be equal to first column of the identity matrix which is 1, 0, 0. So if that’s the case all we have to do is solve these three equations three unknowns and we are not going to show you how to solve the three equations three unknowns because it’s not the best way to solve the inverse of a matrix. But we want to just get to the concept of what it means to have the inverse of a matrix and how to find it.   So if I solve this set of equations by any of the methods which you might have learned so far I will get the four way solution from the matrix from the set of equations being solved I’ll get 0.04762, - 0.9524, 4.571. So this basically tells me - hey this is the first column of the inverse of the matrix. Now we’re going to find out the second column of the inverse matrix so we have 25, 5, 1, 64, 8, 1, 144, 12, 1 and I’m going to put the second column of the inverse matrix here, which is a 1 2 prime, a 2 2 prime, and a 3 2 prime. This is the second column of the inverse matrix and that’ll be equal to the right hand side which is 0, 1, 0, which is the second column of the identity matrix. Again we want to solve these three equations three unknowns and this is what we’re going to get as our solution for our second column of the inverse matrix -0.08333, 1.417 and -5.000. So that’s what I’m going to get as the second column of the inverse of the matrix.   Let’s go and see how we can find the third column of the inverse of the matrix, which we have 25, 5, 1, 64, 8, 1, 144, 12, 1; and now the third column of the inverse matrix would be as follows - would be a 1 3 prime, a 2 3 prime, a 3 3 prime. I’m going to do 0, 0, 1 right here so in order to find the third column of the inverse of this matrix, I will choose the third column of the identity matrix as my right hand side and solve these three equations three unknowns and I’ll be able to find out what my third column of the inverse of the matrix is.   We are not talking about how we would solve these three equations three unknowns  because you can use any method you want to and also this is not the best way to find the inverse of the matrix, but at this time we are just interested in reporting how what it means to find the inverse of a matrix. So in this case we get 0.03571 as the element there,  -0.4643 right here and 1.429 here. So that’s our third column so what we’re going to do now is that we’re going to write down all the three columns which we have obtained so far. So the inverse of the matrix which we have found will be the first column which we found which we got as 0.04762, - 0.9524, 4.571 and then the second column is -0.08333, 1.417 and -5.000 and the third column which we get which we have just written here will be 0.03571, -0.4643, and 1.429. So that’s the inverse of the [A] matrix; so if you would go ahead as an exercise what you can do is take the [A] matrix and the [A] inverse matrix and verify that when you multiply the inverse matrix by [A] that you are going to get the identity matrix. And that’s the end of this segment.