If $A$ is nipotent, how to prove that $A+A^*$ is not nilpotent?

Question

If $A\neq0$ is nipotent, how to prove that $A+A^*$ is not nilpotent?

$A,A^*$ are nilpotent, but I have no idea how to continue

Answer 1

$A + A^*$ is Hermitian, and therefore diagonalizable. A diagonalizable matrix is nilpotent if and only if it is zero.

Suppose, then, that $A + A^* = 0$. That would mean that $A$ is skew-Hermitian. However, a skew-Hermitian matrix is complex-diagonalizable, and is therefore nilpotent if and only if it is zero. So, we conclude that $A = 0$.

Thus, if $A$ is nilpotent and $A+A^*$ is nilpotent, then we must have $A = 0$.