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We know that for a group element $g\in G$, $gh=1$ does not necessarily mean that $hg = 1$. In the case for matrices (linear maps between vector spaces), it is also true that $AB = 1 \nRightarrow BA = 1$. This happens when the $A$ and $B$ are not square matrices (in which case they do not even form a group under multiplication).

However if we restrict the square matrices, $AB = 1 \Rightarrow BA = 1$. What is simple proof of this that avoids chasing the entries, and makes use simply the vector space structure of linear transformations?

(In fact if we could prove this, I think this might imply that for a group to have one-sided(but not two-sided) inverses, it has to be infinite, since every finite group admits a finite dimensional representation.

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