Inverses

Finding an Inverse using an Augmented Matrix

We know that we can use elementary matrices to accomplish the row operations on a matrix

. So, let's suppose that is a square matrix and that we use a succession of elementary matrices to reduce

to the identity matrix. That means we have the following statement:

That means that

This means that if we perform those same elementary row operations on the identity matrix

is by using its cofactor matrix. We already know how to calculate the cofactors .

Then we can show that the inverse matrix of M is the transpose of the cofactor matrix divided by the determinant of M.

This method tends to be a bit more complicated than the method of augmented matrices but it does have some theoretical uses.