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 <a href="https://en.wikipedia.org/wiki/Nakayama_lemma#Proof">Wikipedia</a>), see for example the development given in the Stacks Project <a href="http://stacks.math.columbia.edu/download/algebra.pdf#05G6">here</a>. 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.