True/False question about Hermitian matrices with only real eigenvalues.

Question

Let $A\in \mathbb{C}^{n\times n}$. If all eigenvalues of $A$ are real then $A\in \mathbb{R}^{n\times n}$

I would actually say this is not true because for example:

$$\det \begin{bmatrix}-i-\lambda&1\\\ 1&i-\lambda\\\end{bmatrix} = \lambda^2$$

Is this counterexample enough to say it's False?

Answer 1

This is correct. An easy way to construct lots of counterexamples is to take triangular matrices, with real elements on the diagonal, and non-real elements elsewhere.