## Determinant of a General Circulant Matrix

## Problem 374

Let \[A=\begin{bmatrix}

a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\

a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\

a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\

\vdots & \vdots & \dots & \vdots & \vdots \\

a_{2} & a_3 & \dots & a_{0} & a_{1}\\

a_{1} & a_2 & \dots & a_{n-1} & a_{0}

\end{bmatrix}\]
be a complex $n \times n$ matrix.

Such a matrix is called **circulant** matrix.

Then prove that the determinant of the circulant matrix $A$ is given by

\[\det(A)=\prod_{k=0}^{n-1}(a_0+a_1\zeta^k+a_2 \zeta^{2k}+\cdots+a_{n-1}\zeta^{k(n-1)}),\]
where $\zeta=e^{2 \pi i/n}$ is a primitive $n$-th root of unity.