Can we say that $\text {tr}\ (A) = 0\ $? Let $A$ be an $n \times n$ real matrix with $A^3 + A = 0.$ Can we say that $\text {tr}\ (A) = 0\ $?
I think it's true but can't prove it. Any help will be highly appreciated.
Thanks in advance.

the trace of the matrix is the sum of its eigenvalues, Here  $i,-i,0$ are  the eigenvalues of the given equation $A^3+A=0$ and the sum of its eigenvalues is zero,
So the trace of the matrix is $0$.
A matrix with real entries often has non-real eigenvalues.
