Let $f_A: \mathrm{Mat}_2(\mathbb R)\rightarrow \mathrm{Mat}_2(\mathbb R)$ be such that $f_A(X)=AX$ be an endomorphism of vector spaces. Clearly if $A$ is invertible, then $f_A$ is invertible and ${(f_A)}^{-1}=f_{A^{-1}}$. How can I prove the following statement?
$f_A\;\textrm{invertible} \Rightarrow A\;\textrm{invertible}$