Show that if the eigenvalues of a real matrix are not real, then the matrix cannot be symmetric

Question

Question for the problem

I have not been able to make any progress in this problem and would appreciate any help if possible. I have been able to prove that eigenvalues of a symmetric matrix are real.

Answer 1

Suppose $Av = \lambda v$ with $\lambda $ complex and $A$ real.

Then $x^* Ax = x^* (\lambda x) = \lambda \|x\|^2$ and $(A x)^* x = x^* A^T x= \overline{\lambda} \|x\|^2$.

Hence $x^* (A -A^T) x = (\lambda - \overline{\lambda}) \|x\|^2 \neq 0$. In particular, we must have $A \ne A^T$.