Linear Algebra Proof Help

Hello, I need help with a proof again.

Q: prove that if $\displaystyle A^{2}=A$, then $\displaystyle I-2A=(I-2A)^{-1}$.

So, from that I gather...

Givens: $\displaystyle A^{2}=A \rightarrow A$ is square.

Goal: $\displaystyle I-2A=(I-2A)^{-1}$

So, I don't have very much ha.

Should I just approach this multiplying the the inverse quantity to both sides of the quantity $\displaystyle I-2A$ and hope to get the Identity $\displaystyle I$ back?

Not sure how to use the first fact $\displaystyle A^{2}=A$.

Thanks,

I'll keep plugging away at it.