Cayley-Hamilton/Matrix version/Fact

The theorem of Cayley-Hamilton

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

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

denote the characteristic polynomial of ${}M$.

Then

${}\chi _{M}\,(M)=M^{n}+c_{n-1}M^{n-1}+\cdots +c_{1}M+c_{0}=0\,.$

This means that the matrix annihilates the characteristic polynomial.

Proof, Write another proof