If $m≠n$ and we have the matrices $A$ $(m\times{n})$, $B$ $(n\times{m})$ and $C$ $(n\times{m})$ such that $AB=I(m\times{m})$ and $CA=I(n\times{n})$, does $B=C$?
I know the proof that it is true if we are talking about square matrices, but it doesn't help in this case.
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