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> If $~A~$ is a $~n\times n~$ matrix in $~\mathbb R~$, $~A^3=I~$ and $~AB=-BA~$, how to prove $~\text{tr}(B)=0~$?
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If $~A=I~$, then $~AB=B=-BA=-B \implies B=-B~$. 

Then $~B~$ must be zero matrix with diagonals all zero and must have a trace of zero.

But $~A~$ does not have to equal $~I~$. Am I on the right track and where should I go from here?