For every element v of the vector space consider the pair of vectors (v, Av). Suppose you have A^-1A = I_n.
Since A transforms v to Av, then  A^-1 must transform Av back to v. But then A^-1 followed by A takes Av to v and back again to Av, so that you also have AA^-1 = I_n, at least in its action on vectors that can be written in the form Av.
In finite dimensions every vector is in the range of a non-singular matrix transformation, so that either equation implies the other.