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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?

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

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

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