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?

Hint
$1)$ If $A$ is symmetric then we can write:
$$A=P^TDP$$ 
where $D$ is a diagonal matrix and $P$ is orthogonal. So,
$$A^3=P^TD^3P$$
$2)$ Use that $Trace(XY)=Trace(YX)$
So,
$$Trace(A)=Trace(D)=0$$
and, 
$$Trace(A^3)=Trace(D^3)$$
Now compare $Trace(D)$ and $Trace (D^3)$.
