Counterexample: There exist non-zero nilpotent matrices $A$ and $B$ that $AB$ is non-negative

Question

I am asked to prove or state a counterexample for the following statement:

There exist non-zero nilpotent matrices $A$ and $B$ such that $AB$ is non-negative.

Any ideas would help a lot as I'm not able to come up with a counteraxample. Thanks in advance

Answer 1

Look at

$$A =\begin{pmatrix} 0 & 1\\ 0 & 0\\ \end{pmatrix},\space\space\space B=\begin{pmatrix} 0 & 0\\ 1 & 0\\ \end{pmatrix}.$$