For questions regarding circulant matrices, where each row vector is rotated one element to the right relative to the preceding row vector.
An $n \times n$ circulant matrix is a square matrix which takes the form
$$\begin{bmatrix} c_{0}&c_{{n-1}}&\dots &c_{{2}}&c_{{1}}\\ c_{{1}}&c_{0}&c_{{n-1}}&&c_{{2}}\\ \vdots &c_{{1}}&c_{0}&\ddots &\vdots \\ c_{{n-2}}&&\ddots &\ddots &c_{{n-1}}\\ c_{{n-1}}&c_{{n-2}}&\dots &c_{{1}}&c_{0} \end{bmatrix}.$$
Its determinant is
$$\prod_{j=0}^{n-1}(c_{0}+c_{n-1}\omega^{j}+c_{n-2}\omega^{2j}+\dots +c_{1}\omega^{(n-1)j}),$$
where $\omega$ is the $n$-th root of unity.
