We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
We set $\Phi_n = \left(\omega_n^{(p-1)(q-1)}\right)_{1 \leqslant p,q \leqslant n} \in \mathcal{M}_n(\mathbb{C})$. Justify that $\Phi_n$ is invertible and give without calculation the value of the matrix $\Phi_n^{-1} M_n \Phi_n$.