Let $\mathbb{F}$ be an arbitrary field and $A\in M_{n\times n}(\mathbb{F})$ such that $$tr(A)=0$$ Now show that there exists $P$,$Q$ $\in M_{n\times n}(\mathbb{F})$ such that $$A=PQ-QP$$
It is so natural because we know that $tr(XY-YX)=0$ for all $X,Y$.
I know some proof by induction and canonical forms. Can somebody say another easy and elementary way.