Minimal polynomial/Factorization/Substitutions not bijective/Exercise

Let ${}M$ be an ${}n\times n$-matrix over ${}K$ with the minimal polynomial ${}P\in K[X]$. Let

${}P=F_{1}\cdots F_{k}\,$

be a factorization into polynomials ${}F_{i}$ of positive degree. Show that ${}F_{i}(M)$ is not bijective.

Create a solution