Cayley-Hamilton can be useful in commutative algebra. Related to its close connection with Nakayama's lemma as mentioned in a comment by Qiaochu (see also Wikipedia), see for example the development given in the Stacks Project here. Among the consequences that I find sort of cool and maybe even a little surprising at first glance, we have

• Let $M$ be a finitely generated module over a commutative ring. Then any surjective module map $M \to M$ is an isomorphism.