Q:

Example of non-nilpotent product of nilpotent matrices

What is an example of a positive integer $n$ and two nilpotent $n\times n$ matrices $A$ and $B$ such that $AB$ is NOT nilpotent?

A:

Try $$\begin{pmatrix} 0 & 1 \\ 0& 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 & 0 \\ 1& 0 \end{pmatrix}.$$
Now compute
$$\begin{pmatrix} 0 & 1 \\ 0& 0 \end{pmatrix}\cdot \begin{pmatrix} 0 & 0 \\ 1& 0 \end{pmatrix} =\begin{pmatrix} 1 & 0 \\ 0& 0 \end{pmatrix}   ,$$
which is not nilpotent. 

