Does an identity exist for distributing the inverse for a product including nonsquare matrices?

Question

For example, if $A$ and $B$ are invertible square matrices, we can write $(AB)^{-1} = B^{-1} A^{-1}$.

Now, consider $A$ is an $n \times n$ matrix and $C$ is an $n \times m$ matrix. If $A$ is invertible, does an identity exist for distributing the inverse inside parenthesis of a product of matrices including a nonsquare matrix such as $C$?

For example, if $(C^T A C)^{-1}$ exists, does some identity exist for $(C^T A C)^{-1}$?

Answer 1

A common generalization of an inverse to a non-square matrix $A$ is a Moore–Penrose inverse $A^+$. An equality $(AB)^+=B^+A^+$ holds in special cases, but not in general.