True or False: If A is a square matrix, then all of its eigenvalues are real

Question

This is one true or false question from a past exam of my Linear Algebra course:

If $A$ is a square matrix of real entries, then all of its eigenvalues will be real numbers different to zero. Mark True or False.

So, I know that if it is a symmetrical matrix, all of its eigenvalues will be real (but they don't specify that) so... In my reasoning, as you have to get a characteristic polynomial in order to find the eigenvalues, it is possible that one of its solutions come to be complex. So I think it is false but I'm not really sure and also I don't know how to prove it.

Answer 1

Just provide a concrete example. If, say,$$A=\begin{bmatrix}0&-1\\1&0\end{bmatrix},$$then $A$ is a real matrix, but its eigenvalues are $\pm i$.