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?

You want two matrices $A,B \in M(n\times n;K)$ such that $AB=0$ and $BA\neq 0$. Define
$$A= \begin{pmatrix}1 &1\\0&0\end{pmatrix},\quad B=\begin{pmatrix}-1&0\\1&0\end{pmatrix}$$
two matrices in $M(2\times 2; \Bbb{R})$ You can see that 
$$AB = \begin{pmatrix}0&0\\0&0\end{pmatrix}$$
while
$$BA = \begin{pmatrix}-1&-1\\1&1\end{pmatrix}$$
As you can see matrix multiplication is not commutative.
