Is this a sufficient proof for showing that if matrices commute, their inverses commute?

Question

If not, where am I going wrong, or am I assuming something that isn't true?

$AB=BA$, then $\left(B^{-1}A^{-1}\right)^{-1}=\left(A^{-1}B^{-1}\right)^{-1}$ so $B^{-1}A^{-1}=A^{-1}B^{-1}$.