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

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  • $\begingroup$ 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. $\endgroup$ Commented Oct 20, 2020 at 15:01

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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.

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  • $\begingroup$ I don't see how this can be useful for deriving an identity. The pseudoinverse is not equal to the inverse. $\endgroup$
    – Ralff
    Commented Oct 19, 2020 at 2:04
  • $\begingroup$ @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. $\endgroup$ Commented Oct 19, 2020 at 8:12

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