




For every element v of the vector space consider the pair of vectors
(v, Av). Suppose you have A^{1}A = 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 nonsingular matrix transformation,
so that either equation implies the other.