$AB$ and $BA$ do not have the same eigenvalue 
Suppose $A$ is $n\times m$ and $B$ is $m\times n$ where $n\not = m$.
  Show $AB$ and $BA$ may not have the same eigenvalues.

Can someone offer me a simple counterexample for this claim? Is it required to prove the statement for all $n,m$, or can a counterexample be used?

Hint: They actually have the same nonzero eigenvalues.  Look for a small example where one of $AB$ and $BA$ is invertible.  The other will not be invertible.
