Does $\operatorname{tr} (A)=0$ imply $\operatorname{tr} (A^3)=0$? Let $A_{(2n+1)\times(2n+1)}$ be a symmetric matrix of Rank $2n$. Then does $\operatorname{tr}A=0$ imply $\operatorname{tr}A^3=0$? If not, Under what condition?

The answer is NO. For a counterexample, let 
$$A=\operatorname{diag}(1,3,-2,-2,0)$$
We have $\operatorname{Tr}(A)=0$ and $\operatorname{Tr}(A^3)=12\ne0$.
