We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the characteristic polynomial of $Z_n$ is $X(X^{n-1} - 2)$.
Deduce that $Z_n$ is imprimitiv and specify its coefficient of imprimitivity.