Simplification of equation with matrices and inverses

Question

I tested the following simplification numerically and it doesn't seem to be working:

$$ A^{-1} B^{-1} A B = I $$

Why is this invalid? Isn't the left side equivalent to $ (AB)^{-1} (AB) $? A and B are invertible square matrices.

Answer 1

$$(AB)^{-1} = B^{-1} A^{-1} \ne A^{-1} B^{-1}.$$

Answer 2

Let $(AB)^{-1}=X$.

Thus, $$ABX=I$$ or $$A^{-1}ABX=A^{-1}I$$ or $$BX=A^{-1}$$ or $$B^{-1}BX=B^{-1}A^{-1}$$ or $$X=B^{-1}A^{-1},$$ which says $$(AB)^{-1}=B^{-1}A^{-1}.$$