Show that if a is any m x n matrix, then ImA = A and AIn = A

Homework Statement

Show that if A is any m x n matrix, then the m x m identity multiplied by A = A, and A multiplied by the n x n identity = A.

The Attempt at a Solution

I know how to prove this by writing out a general m x n matrix, and multiplying it by the identity, but is there a better way of showing this? It just seems kind of silly when I write it out...

$$(AI)_{i,j } = \sum_{r= 1}^n A_{i, r} I_{r, j} = a_{i, j}$$