CHAPTER 05.24: SYSTEM OF EQUATIONS: Finding the inverse of a matrix by adjoints Example     In this segment we will take the inverse of a matrix by using the adjoints method, let’s take an example here. Somebody asks you to find the inverse of this matrix here: 25, 5, 1, 64, 8, 1, 144, 12, and 1. So find the inverse of this matrix. That is the problem statement. So we know that in order to find the A inverse-what we have to do is, we have to find the determinate of the A matrix and then we also have to find the adjoint of A. If we are able to do that, then we are able to find the inverse of the matrix. So let’s look at a few steps. The determinate of A is given to you as -84. We have done this in a previous segment. Let’s go and see how to find the adjoint of A. As we said, adjoint of A is going to be based on theco factors.   Let’s see, we are going to find cofactors C11. How do we find cofactor C11? Cofactor C11 will be corresponding to the element A11, which is 25, right here. That is defined as -1 raised I plus 1 times M11- well 1 plus 1 I should say-times M11. M11 is the minor of the matrix corresponding to A11. So C11 is the first row and first element of the cofactor matrix and that’s the equal to -1 raised to the power of I plus J. I plus J. M11 is the minor of the –corresponding to the A11 element. So this gives us M11 itself. Now how do I find M11? I find M11, the minor of the entry M11 by simply getting rid of the first row and the first column. So get rid of the first row and first column. So I get rid of the first column and the first row. First row and first column. I am left with 8, 1, 12 and 1. And then I find the determinate of whatever I am left with. I am left with the sub-matrix of 2 by 2. 8, 1, 12, and 1. Finding the determinate of a 2 by 2 is pretty simple. Its 8 times 1 minus 12 times 1 and that gives you -4. So that is how I found out the C11, which is equal to M11, which is equal to -4. This gives me C11is just -4. Let’s go and find out another cofactor corresponding to, let’s suppose, C12 to A12. Let’s go ahead and find what C12 is. C12 would be -1 raised power I plus J, which would be 1 plus 2. M12 and that would be –M12.   Now what is M12? M12 will be the minor of the matrix corresponding to the A12 element. But it will be the determinate of the matrix that is left over, once I take the 1st row out and the 2nd column out. So if I take the 1st row out and the 2nd column out, I am left with: 64, 1, 144, and 1. That is nothing more than 64 times 1 minus 144 times 1. That is -80. So that is what M12 is, so that means that C12 is equal to –M12. Since M12 is -80, C12 is going to be 80. So you get the point on how you are going to calculate the cofactor matrix corresponding to each of the 9 elements, which you have in the A matrix. We found C11 equal to -4. We found C12 equal to 80. We can similarly find C13 to be -384 you can (4:22) C21 turns out to be 7. C22 turns out to be -119. C23 turns out to be 420. Then C31 turns out to be -3, C32 is 39. C33 turns out to be-120. These are the cofactors which we get, corresponding to the 9 elements of the A matrix. So now I can write my C matrix, which is my cofactor matrix. So I just write it in the matrix form. -4, 80, -384. It’s simply putting these elements in the proper place. 7, -119, 420, -3, 39, and -120.   In order to find out the inverse of the matrix, all I have to do is dividing 1 by the determinate of the matrix, which is 84 times the transpose of this matrix. So it will be the transpose of this matrix. It is a C transpose. Let me write this down. A inverse is the adjoint of A divided by the determinate of A and the adjoint of A is nothing more than the transpose of the cofactor matrix. So the transpose of this matrix is -4, 7, -3, 80, -119, 39, -384, 420, and -120. That is the transpose of this matrix. So that is C transpose. And now I divide it by the determinate of the matrix. And if I do this division I’ll be able to find out what the inverse of the matrix is, which turns out to be- The determinate of the matrix is -84- and it turns out to be 0.04672, -0.08333, 0.03571, 0.9524, 1.147, -0.4643, 4.571, -5.000, and 1.429. And that is how we are able to find the inverse of a matrix. You can extend this procedure to a 4 by 4 or 5 by 5 but the hand computations are going to be lengthy, even if you were going to use a computer and follow the algorithm here, that is still going to take a lot of time. And this is the end of this segment.