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

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 $$(C^T A C)^{-1}$$?

• It requires the general Moore-Penrose pseudoinverse of a matrix, and that too only in certain cases. Basically, if $C$ is non-square, the only way to work with $C$ is using the notion of inverse for a non-square matrix , which extends the usual notion of an inverse. Indeed, the MP pseudoinverse must come into play, as the answer below indicates. Commented Oct 20, 2020 at 15:01

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.
• @Ralff I don’t understand. When a matrix $A$ is invertible, its pseudoinverse is its inverse, see, for instance, here. When $A$ is not invertible, its inverse does not exists. Commented Oct 19, 2020 at 8:12