I guess this is the way you need to go...
However I found this- adjoining the matrix A with the identity matrix I: [ A I ] then use elementary row options to get the form to [ I A-1 ]
"The process shown in Example 3 applies to any n x n matrix A and will find the inverse of A, if it exists." - From the book in regards to the method above.
This is called "finding the inverse of a matrix by Guass-Jordan Elimination". I believe this is the way I need to prove this. All other theorems in my text book involve bringing in a second matrix B (which is assumed to be the inverse of A).