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Precisely, why is multiplicative identity defined to be $IA=AI=A$ why both sides should work why not something like $AI=A$ ? Is there an underlying advantage?

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  • $\begingroup$ If matrices are over a field, then indeed the condition $$\forall A \quad AI = A$$ gives a unique solution $I=\sum_j e_j\otimes e_j$. However, if the underlying structure is not a field, I would suppose that only one equality is not sufficient (I'll try to sketch a counterexample). $\endgroup$ Jul 16, 2019 at 9:01
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    $\begingroup$ If $A$ is non square, if $AI$ is defined, the product $IA$ is not, that's all. $\endgroup$
    – Bernard
    Jul 16, 2019 at 9:06
  • $\begingroup$ @Bernard , yes but that itself is my question I'm asking why is the rule defined like it has been ? $\endgroup$ Jul 16, 2019 at 9:11
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    $\begingroup$ Note that $IA=A=AI$ should be written $I_mA=A=AI_n$ when $A$ is an $m\times n$ matrix. The identity matrices on the left and the right are not the same. $\endgroup$ Jul 16, 2019 at 9:56
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    $\begingroup$ For $m\times n$ matrices one can define a "left identity" via $IA=A$ and a "right identity" via $AI=A$. But these are different, and neither of them is itself an $m\times n$ matrix. $\endgroup$
    – user856
    Jul 16, 2019 at 9:59

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The use of category theory may make this clearer. Suppose we have a category of objects $\, V_1, V_2, V_3, \dots\,$ An $\,n \times m\,$ matrix $\,A\,$ is an arrow from $\,V_m\,$ to $\, V_n.\,$ Matrix multiplication is only defined between compatible matrices. That is, If $\,B\,$ is an arrow from $\,V_n\,$ to $\,V_k\,$ then the matrix product $\,B A\,$ is an arrow from $\,V_m\,$ to $\,V_k.\,$ Each object $\,V_n$ has a unique identity arrow denoted by $\,I_n\,$ which is the $\,n \times n\,$ identity matrix. This gives us the identity $\, A = AI_m = I_nA.\,$ The two identity arrows are not the same unless $\,n=m.\,$

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  • $\begingroup$ I honestly don't see how the use of category theory makes this any "clearer" than the bare linear algebra facts would... $\endgroup$
    – user856
    Jul 16, 2019 at 18:28

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