I'm trying to prove this:
Let $R$ be a ring, and $A \in M_n(R)$. Write $L_A$ for the linear map $L_A : R^n \to R^n$ determined by left multiplication by $A$. Shows that if $L_A$ is injective, then $\det(A)$ is not a zero divisor.
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You meant a commutative ring. The adjugate matrix should help. – reuns Nov 14 '18 at 01:58
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1Particular case (for $m=n$) of part b) of https://artofproblemsolving.com/community/c7h124137 . – darij grinberg Nov 14 '18 at 02:19
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Thanks! Works for me. – Jeremiah Goertz Nov 15 '18 at 00:30
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https://math.stackexchange.com/q/161523/469000 – Unknown Nov 14 '18 at 01:57