If $A$ and $B$ are matrices, what can we conclude if $AB = BA$

Question

Assume $A$ and $B$ are $n$ by $n$ matrices. If we know $AB$ = $BA$, what can we conclude from that?

I can think of three cases so far:

1. $A=B$: By Substitution on $A\times A=A\times A$, we get $AB=BA$
2. $A=I$ or $B=I$: If $B=I$, since $AI=IA=A$, $AB=BA$
3. $A$ or $B$ are filled with only zeros: A zero matrix multiplied by any other matrix in either direction is always a zero matrix.

Are there any cases I am missing? Any other information I can get out of this?