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: 
$A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 0 & 0 \\ 1 & 0\end{bmatrix}.$ When thinking of examples try to think of most trivial ones like $O$, $I\_n$ etc. These are also kind of trivial right?
We can think of constructing such matrices by using the Rank-Nullity theorem. Note that one matrix should be singular by taking the determinant on both sides. Now choose columns arbitrarily in the singular matrix $A$ such that the column space doesn't equal dimension of full space, i.e, choose at least one less linearly independent column than the order of matrix. But we must choose atleast one non-zero vector. For the other matrix $B$ choose columns which are exactly in the null space of the first matrix. Then, we would get $AB=O$, whereas by virtue of there being some linearly independent vectors in $A$, we would get $BA\neq O$. I think taking examples in second order matrices and generalizing using block matrices would as Andreas answer has done it above.