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Let $A\in M_{m\times n}(\mathbb C)$ and $B\in M_{n\times m}(\mathbb C)$ be two matrices. If $AB$ is invertible, does it mean that $A$ and $B$ are invertible? If so, then how do I prove that? Thanks in advance!

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    $\begingroup$ How do you define the inverse of a $m×n$ matrix if $m\neq n$? $\endgroup$ – P. Siehr Nov 8 '17 at 8:35
  • $\begingroup$ To clarify, my question was not meant to be offending in any way. Rather: Do you also think about pseudo inverses in your question. $\endgroup$ – P. Siehr Nov 8 '17 at 8:44
  • $\begingroup$ No, $A$ and $B$ will not be invertible in general but you can prove that $A$ has right inverse and $B$ has left inverse. $\endgroup$ – A---B Nov 8 '17 at 8:50
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No.

To be invertible, a matrix has to be a square matrix which is not the case here.

If they are indeed square matrices, note that $\det(AB)=\det(A)\det(B)$. Try to prove it using this identity.

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  • $\begingroup$ Is there a solvation without using determinants? we've just started this chapter $\endgroup$ – Anonymus Nov 8 '17 at 9:12
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    $\begingroup$ If $(AB)C = I$, then $A(BC)=I$, try to prove $B$ is invertible too for the context of square matrix. $\endgroup$ – Siong Thye Goh Nov 8 '17 at 9:19

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