commutative matrix multiplication of nxn matrices? If there are two matrices A and B that are both nxn matrices, will AB = BA always?
Is there a way to have those two matrices so that AB = 0 but BA ≠ 0?

No. I can give a simple counterexample:
$\begin{pmatrix}1 & 2 \\ 3 &4\end{pmatrix}$ * $\begin{pmatrix}4 & 3\\2 & 1\end{pmatrix}$ = $\begin{pmatrix}8 & 5\\20 & 13\end{pmatrix}$
while
$\begin{pmatrix}4 & 3\\2 & 1\end{pmatrix}$ * $\begin{pmatrix}1 & 2\\3&4\end{pmatrix}$ = $\begin{pmatrix}16 & 20\\8 & 8\end{pmatrix}$.
Two matrices are rarely commutative. Even square ones.
Check this question here for more on commutativity with matrices.
