Linear algebra with a transformation i think

Question

Let $V$ be a finite-dimensional real vector space and let $P:V→V$ be a linear map such that $P^2 = P$. Which of the following must be true?

1. $P$ is invertible
2. $P$ is diagonalizable
3. $P$ is either the identity map or the zero map.

What can be said about the possible eigenvalues of $P$?

Any help will be much appreciated

Answer 1

If $P$ is of the Jordan Canonical Form, then it is easy to verify that $P^2=P$ implies (2). In addition, you can verify that $P^2=P$ if and only if $P$ is a diagonal matrix with eigenvalues in $\{0,1\}$.