Cayley-Hamilton/Matrix version/Fact

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

${\displaystyle {}\chi _{M}=X^{n}+c_{n-1}X^{n-1}+\cdots +c_{1}X+c_{0}\,}$

denote the characteristic polynomial of ${\displaystyle {}M}$.

This means that the matrix annihilates the characteristic polynomial.