Let $M$ be an $n\times n$ diagonalizable matrix with characteristic polynomial, find the matrix

Question

Let $M$ be an $n\times n$ diagonalizable matrix with characteristic polynomial

$a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$,

Find the matrix

$a_nM^n+a_{n-1}M^{n-1}+...+a_1M+a_oI$,

where $I$ is the $n\times n$ identity matix.

Please give me some hints.

Answer 1

If $p( \lambda )$ is the characteristic polynomial of $A$ then $p(A)=0$ for the Hamilton Cayley theorem