I know the proof in terms of linear algebra where you simply argue that if ${AB = I}$ then ${A}$ must be bijective, so since ${B}$ is the inverse map on one side, it must be an inverse on the other side, so ${BA}$ = ${I}$. But I'm interested if there's any way purely going from the identity ${A_{ij}B_{jk}} = {\delta_{ik}}$ to ${{B_{ij}A_{jk}} = \delta_{ik}}$ using algebra and no lifting to a more abstract context. I'm aware of the answers given here: If $AB = I$ then $BA = I$, but all of these use more abstract properties of matrices. These equations so simple it seems like there should be a nice way of proving them without requiring any constructs on top of the basic definitions.
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Since there are algebraic systems in which a left inverse need not be a right inverse you must necessarily use something from the linear algebra context to prove the theorem. – Ethan Bolker Aug 12 '23 at 14:36
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1@EthanBolker I'm willing to use the algebraic definition of matrix multiplication. – latbbltes Aug 12 '23 at 14:37
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@latbbltes, Here is my answer. Tell me if you like it. – Angelo Aug 12 '23 at 14:42
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In the proposed duplicate question, this answer in particular satisfies your criterion I believe. – Dustan Levenstein Aug 12 '23 at 14:50
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1https://math.stackexchange.com/a/3895/18966 – Dustan Levenstein Aug 12 '23 at 14:50
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@DustanLevenstein Love it, thank you – latbbltes Aug 12 '23 at 14:52
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"but all of these use more abstract properties of matrices": no, some were extremely elementary. – Anne Bauval Aug 12 '23 at 16:24