Question: I need to construct square matrices $A$ and $B$ such that $AB=0$ but $BA \neq 0$. 

I know matrix multiplication is not commutative, but I don't know how to construct such matrices. Thanks in advance.

Edit: looking for some simple way

Answer: 
A whole family:
$$ A=\begin{bmatrix}0&a\\0&a'\end{bmatrix}, \quad B=\begin{bmatrix}b&b'\\0&0\end{bmatrix}$$
with non-orthogonal vectors $(a,a')$ and $(b, b')$.
Yet another more powerful (and maybe more "advanced"):

Let A be a given matrix, now $AB-BA$ can be rewritten with help of matrix multiplication and vectorization as: $M\_{AR}\text{vec}(B) - M\_{AL}\text{vec}(B)$, where $M\_{AR}$ is a matrix performing multiplication from the Right by $A$ and $M\_{AL}$ from Left by $A$.

We can now first pick a vectorization $\text{vec}$ and an objective matrix $C$ and try to solve $$\min\_B\{\|(M\_{AR}-M\_{AL})\text{vec}(B)-\text{vec}(C)\|\}$$

Where $C = AB-BA$ is something we can choose.