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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.

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    @Listing: I don't really consider that as a full correct answer, but on your insistence I've posted it.2011-11-14

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Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0\end{bmatrix}$ and let $B = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$.