Inverse of matrix product

Question

Let A be an $n\times n$ matrix and B be an $n\times m$ matrix with $m<n$ can I use this identity?

$(AB)^{+}=B^+A^{-1}$

I am not sure that this is the right inverse of the product $AB$ if this is correct and works as the inverse of square matrices product can you give me a reference? Otherwise I need ideas. Thanks

Answer 1

I suppose that you are asking whether $(AB)^{\color{red}{+}}=B^+A^{-1}$ rather than whether $(AB)^{\color{red}{-1}}=B^+A^{-1}$. As other users have pointed out $AB$ is not a square matrix in general and it makes no sense to talk about $(AB)^{-1}$.

Is $(AB)^+=B^+A^{-1}$? Unfortunately, the answer is no. Out of the four defining properties of Moore-Penrose pseudoinverse, $B^+A^{-1}$ only satisfies three. In particular, $(AB)(AB)^+$ should be Hermitian, but that in general does not hold if you put $B^+A^{-1}$ in place of $(AB)^+$.

Answer 2

No, you cannot do that, since $AB$ is not a square matrix and therefore it has no inverse.