Here is one way that I might do it.
You are assuming:
So multiply both sides by the matrix A which gives you:
Now add the identity matrix to both sides of the equation:
Observe that the bit in parantheses is the expression we assumed is exactly equal to the matrix of all 0's. Sustitute using that fact to arrive at:
The matrix is just which is really or .
So not only have you shown that the inverse exists but also that the inverse of is in fact since which is the definition of an inverse matrix.