Given P idempotent, show that I−P is idempotent.

Question

So my task is well summed up by this older post:

Given $P$ idempotent, show that $I-P$ is idempotent.

PandaMan idea is that by proving $(I−P)^2 = (I-P)$ we prove that $(I-P)$ which implies that also $(I−P)^2$

My question is, how do we know/prove that $(I−P)^2$ is idempotent?

Thanks in advance! /Luke

Answer 1

$$(I-P)^2 = I - 2P + P^2 = (I - P) + (P^2-P) = I-P$$

because $P^2-P = 0$ when $P$ is idempotent.