I have follow matrix equation ($A$ and $B$ are $N\times N$ matrices over some field)
$AB=BA=0$
with
$A\ne 0$
$\det{A}=0$
$\operatorname{Tr}{A}=0$
I suppose that
$\operatorname{Tr}{B}=0$
Is it so? And if it's so then what is the idea of the proof?
Thank you.