"Prove or give a counterexample: If A, B, C are 2x2 matrices, AB = BC, and B is invertible, then A = C."

I'm greatly lost on how to solve this problem; I've been trying for a very long time. I've played around with numbers enough to convince myself that this is indeed a proof. However, I cannot figure out how to prove it unless A,C are either the identity matrix, the zero matrix, or the inverse of B (or if B is the identity matrix). Any help is appreciated.

EDIT:

Turns out there are counterexamples. Solutions were distributed after assignment was turned in :\ .