Let $\zeta$ be a primitive $n$-th root of unity.
Prove that all eigenvalues of the matrix $\left(\begin{matrix} 1 & 1 & 1 & \cdots & 1\\ 1 & \zeta & \zeta^2 & \cdots & \zeta^{n-1}\\ 1 & \zeta^2 & \zeta^4 & \cdots & \zeta^{2(n-1)}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & \zeta^{n-1} & \zeta^{2(n-1)} & \cdots & \zeta^{(n-1)^2} \end{matrix}\right)$ have modulus $\sqrt{n}$.