Let be a square matrix. We now have a method of determining whether or not is invertible: do row operations to until you reach a matrix in RREF. Then by Theorem 3.12.2 is invertible if and only if the RREF matrix is the identity.
What if we actually want to know what the inverse matrix is? You probably already know that a matrix is invertible if and only if , and in this case
This formula does generalise to larger matrices, but not in a way which is easy to use: for example, the general formula for the inverse of a invertible matrix is
where means and
This isn’t a formula that you want to use. Luckily we can use RREF techniques to determine invertibility and find inverses.
Let be an matrix, and suppose we want to find out whether is invertible and if so what its inverse is. Let be the identity matrix. Here is a method:
Form the super-augmented matrix .
Do row operations to put this into RREF.
If you get then is invertible with inverse .
If the first part of the matrix isn’t then isn’t invertible.
It works because the first part of the matrix is a RREF matrix resulting from doing row operations to , so if it is then by Theorem 3.12.2 is invertible, and if it is not then is not invertible. It just remains to explain why, in the case is invertible, you end up with .
Think about the columns of the inverse of . We have , so , , etc, where is the ith column of . So is the unique solution of the matrix equation . You find that by putting into RREF, and you must get since is the only solution.
Repeating that argument for every column, when we put into RREF we get , that is, .
Let . To find whether is invertible, and if so what its inverse is, we put into RREF:
This is in RREF, so the inverse of is
as you can check by multiplying them together.