Permutation matrix/Cycle/Characteristic polynomial/Factorizaton/Sum decomposition/Exercise

Let ${}\pi$ be a cycle of length ${}n$, and let ${}M$ denote the corresponding permutation matrix, that is,

${}M={\begin{pmatrix}0&0&0&\ldots &1\\1&0&0&\ldots &0\\0&1&0&\ldots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&0&\ldots &1&0\end{pmatrix}}\,.$

1. Determine the characteristic polynomial ${}\chi _{M}$ of ${}M$.
2. Show that ${}P=X-1$ divides ${}\chi _{M}$, and compute the factorization

   ${}\chi _{M}=PQ\,.$

3. Determine ${}P(M)$ and ${}Q(M)$.
4. Determine ${}\operatorname {kern} P(M)$ and ${}\operatorname {kern} Q(M)$.

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