Suppose that $X$ and $Y$ are both $n \times k$ matrices with $n > k$.
Are there any matrices that satisfy $YX' = I_n$?
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The rank of $YX'$ is less than the rank of $Y$ and the rank of $X$, both of which are at most $k$. The identity has rank $n >k$ which would violate $YX' = I$.