Prove that if matrix $~A^2 = I~$, then $~A~$ is diagonalizable.

Question

I have to prove that for $A \in M_n(\mathbb{C} ) $ , if $A^2 = I$, then $A$ is diagonalizable.

I have no direction what so ever, Please help.

Answer 1

$A$ is a root of $x^{2} - 1$, which has distinct roots. Now the minimal polynomial of $A$ must divide $x^{2} - 1$, and so it has distinct roots too. This implies that $A$ is diagonalizable.

Answer 2

Hint: Either $A= \pm I$ or $X^2-1$ is the minimal polynomial of $A$.