CHAPTER 05.22: SYSTEM OF EQUATIONS: Finding the inverse of a matrix by adjoints Theory

In this segment we will talk about how the inverse of a matrix by using adjoints. For small matrices like 3 by 3 or 4 by 4 matrices, if you are trying to find the inverse by hand, this might be the easiest way to do it. But again, this particular method is not encouraged especially for large order of matrices because computation time- we are not talking about doing it by hand but if you are doing it by computer, that the amount of time it would take to find the inverse matrix by this method would be enormous. Just for the sake of completion we do want to talk about this method because there are many people who still use this method to find the inverse of a matrix. So the (55) we are going to define as simply finding the adjoint of A which is this matrix you are trying to find the inverse, divided by the determinate of the A matrix. So what does it mean to-adjoint of A. Adjoint of A is nothing but the transpose of the cofactors. So we know that, if want to look at the cofactors of A they correspond to each of the elements. So each of the elements has a cofactor. So if we find out the cofactor for each of the elements and then we gave the transpose of that - that gives us the adjoint of A. All we have to do is take Adjoint of A and divide it by this number, which is the determinate of A. We should be able to find out what the inverse of the matrix is. S CIJ is the cofactor of AIJ. That’s what we mean by each of these elements, which are in the cofactor matrix right here. And that is the end of this segment.