matrices - Proof for linear algebra theorem

I want to prove that if $AB=cI$ for some matrices $A,\ B$ and number $c$, then $AB = BA$.
I start my proof with $c \neq 0$. then, $A$ and $B$ are invertible, And $BA = B(cB^{-1}) = cBB^{-1} = cI = AB.$
What about $c = 0$?
I guess it's true, but i'm not sure. If I true for $c = 0$, what is the proof? If It's not, what is the contradiction example?
Thank you.