Prove that if A is idempotent and fullfills $A = A^{-1}$ then it follows that $A = I_n$

Question

I'm currently learning linear algebra and I have stumbled across the following example in my book without a solution.

A matrix $B \in \mathbb {R}^{n x n} $ is called idempotent if $BB = B$. Prove that if A is idempotent and fullfills $A = A^{-1}$ then it follows that $A = I_n$.

Could someone please help me?

Answer 1

$A = A^{-1}$ gives $A^2=I_n$. Since $A$ is idempotent, we have $A=A^2.$

Answer 2

Hint: $AA^{-1}=I_n$ by definition.