Solve Idempotent Matrix Inequality: n≥p-1 | Artin's Algebra

[Dunkle]

Homework Statement

This is from Chapter 11 of Artin's Algebra:

Let p be a prime, and let A (not the identity) be an nxn integer matrix such that $$A^{p}=I$$. Prove that $$n \geq p-1.$$

Homework Equations

This is in the factorization chapter, and the section is called Explicit Factorization of Polynomials.

The Attempt at a Solution

I don't even know where to begin. I'm guessing I need to somehow get a polynomial equation, but like I said, I don't really know where to start. Any help would be greatly appreciated!

 

[lanedance]

haven't got there yet, but hopefully these help you get started

clearly A is invertible as A^(p-1) = A^-1

also note that
A^(p+1) - A = A(A^p - I ) = 0