Determinant of $ n \times n$ matrix and its characteristic polynomial. Suppose, $M_4, M_5,..M_n$ is as follows then determinant and characteristic polynomial of $M_n$. $M_4=\left(
\begin{array}{cccc}
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1 \\
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
\end{array}
\right),M_5=\left(
\begin{array}{ccccc}
 0 & 0 & 1 & 1 & 0 \\
 0 & 0 & 0 & 1 & 1 \\
 1 & 0 & 0 & 0 & 1 \\
 1 & 1 & 0 & 0 & 0 \\
 0 & 1 & 1 & 0 & 0 \\
\end{array}
\right),M_6=\left(
\begin{array}{cccccc}
 0 & 0 & 1 & 1 & 1 & 0 \\
 0 & 0 & 0 & 1 & 1 & 1 \\
 1 & 0 & 0 & 0 & 1 & 1 \\
 1 & 1 & 0 & 0 & 0 & 1 \\
 1 & 1 & 1 & 0 & 0 & 0 \\
 0 & 1 & 1 & 1 & 0 & 0 \\
\end{array}
\right), \quad M_8=\left(
\begin{array}{cccccccc}
 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\
 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\
 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\
 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\
 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\
 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
\end{array}
\right)$

The matrix $M_n$ is a circulant matrix where $c_0=c_1=c_{n-1}=0$ and other $c_k=1$. Eigenvalues and eigenvectors of circulant matrices can be readily calculated.
