Let $\displaystyle A$ be an invertible matrix. Prove that if $\displaystyle A$ is diagonalizable, so is $\displaystyle A^{-1}$.

I'm afraid I don't quite know how this could be proven. This is the closest thing I have: proving that if $\displaystyle A$ is diagonalizable, so is $\displaystyle A^T$.

$\displaystyle P^{-1}AP=D\Rightarrow P^TA^T(P^T)^{-1}=D$ Since $\displaystyle D=D^T=(P^{-1}AP)^T=P^TA^T(P^T)^{-1}$

I doubt that proof is even relevant to the question. Does anyone know how to prove it?