Unique A^-1

This is (hopefully) the same answer you got for p by trial and error a few pages ago. If A is non-singular (has an inverse) and Ap = q, then p = A^-1q.

The inverse of a non-singular square matrix is unique. One way to see this is that there is only one column matrix p that is the solution to Ap = q, so there must be only one A^-1.

It might look like computing A^-1 is a useful thing to do. In fact, A^-1 is more useful in discussions about matrices and transformations than it is in actual practice. Almost never do you really want to compute a matrix inverse.

For example, say that a column matrix p represents a point in a computer graphic world. The viewpoint changes, and the column matrix is transformed to q = Ap. If you want to talk about reversing the transformation, you talk about A^-1q. But almost always there is an easier way to reverse the transformation than to compute the inverse.