If A^2 = I for a matrix A, then A is diagonalizable

So I'm at Walmart checking out the prices of frozen pizza, and the cute older woman is next to me. She turns to me, smiles, and says "Is it true of A^2 is the identity matrix, that it's also diagonalizable?". Ok so that wasn't funny and that's not how it goes, but this is a VERY useful fact that just comes up so often, at least in the world of mathematics, that it makes it worth mentioning. Here we go.

If A is a matrix such that A^2 = I, then A^2 - I = 0. Set f(x) = x^2 - 1. Then f(A) = 0 by Cayley Hamilton, so the minimal polynomial of A, say p, divides f. In particular this means that p has no repeated factors, hence A is diagonalizable.