Matrix/Annihilating polynomial/Inverse matrix/Exercise

Let ${}M$ be an ${}n\times n$-matrix over a field ${}K$, and let

${}P=a_{0}+a_{1}X+\cdots +a_{m}X^{m}\in K[X]\,$

be a polynomial with

${}P(M)=0\,$

and with ${}a_{0}\neq 0$. Show that ${}M$ is invertible, and that its inverse matrix is given by

${}M^{-1}=-{\frac {1}{a_{0}}}{\left(a_{1}+a_{2}M+\cdots +a_{m}M^{m-1}\right)}\,.$

Create a solution