[Math] Can a non-zero symmetric matrix have only zero eigenvalues

Question

I know that a non-zero symmetric 2×2 matrix can't have only zero eigenvalues ( a zero eigenvalue with algebraic multiplicity 2), since such a matrix should have complex off diagonal entries to satisfy both trace and determinant being zero. I'm guessing if this is the case for the general case of any non-zero n×n symmetric matrix.

Answer 1

No, because then it is a nilpotent symmetric matrix, and since symmetric matrics are diagonalizable and the only diagonalizable nilpotent matrix is the zero matrix, we'd get a contradicition to "non-zero matrix"