A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a $3 \times 3$ circulant matrix. $$\begin{bmatrix} 1 & 2 & 3\\ 3 & 1 & 2\\ 2 & 3 & 1 \end{bmatrix}$$For any $n \times n$ circulant matrix ($n > 5$), which of the following $n$-length vectors is always an eigenvector?
(a) A vector whose $k$-th element is $k$
(b) A vector whose $k$-th element is $n k$
(c) A vector whose $k$-th element is $\exp \left(j\dfrac{2π(n − 5)k}{ n}\right)$ where $j = \sqrt{−1}$
(d) A vector whose $k$-th element is $\sinh (2πk/n)$
(e) None of the above
I don't know what to start with. Can someone please help?