If $\omega$ is non real cube roots of unity , then the eigenvalues of the matrix

Question

If $\omega$ is a non-real cube root of unity, then what are the eigenvalues of the matrix

$$\left[ \begin{matrix} 1 & 1 & 1\\ 1 & \omega & \omega^{2} \\ 1 & \omega^{2} & \omega^{4} \end{matrix}\right] \left[ \begin{matrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 1 & 0 & 0 \end{matrix}\right] \left[ \begin{matrix} 1 & 1 & 1\\ 1 & \frac{1}{\omega} & \frac{1}{\omega^{2}}\\ 1 & \frac{1}{\omega^{2}} & \frac{1}{\omega^{4}}\end{matrix}\right]?$$

Answer 1

Hint: The matrix is of the form $B=PAQ$. Note that $PQ=3I$. Therefore, $B=P(3A)P^{-1}$ and so $B$ and $3A$ have the same eigenvalues.