A Proof of the Cayley-Hamilton Theorem

Let M (n, n) be the set of all n × n matrices over a commutative ring with identity. Then the Cayley Hamilton Theorem states: Theorem. Let A ∈ M (n, n) with characteristic polynomial det(tI − A) = c 0 t n + c 1 t n−1 + c 2 t n−2 + · · · + c n. Then c 0 A n + c 1 A n−1 + c 2 A n−2 + · · · + c n I = 0. In this note we give a variation on a standard proof (see [1], for example). The idea is to use formal power series to slightly simplify the argument. Proof. First, observe that det (I − tA) = t n det 1 t I − A = c 0 + c 1 t + c 2 t 2 + · · · + c n t n .