Cayley-Hamilton theorem

Theorem. (Cayley-Hamilton) Let $T$ be a $k$-linear endomorphism of a finite-dimensional vector space $V$ over a field $k$. Let $P_T(x)$ be the characteristic polynomial

$\displaystyle P_T(x)=\det(x\cdot 1_V-T)$

Then

$\displaystyle P_T(T)=0\in\mathrm{End}_k(V)$.