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I am looking at proofs of the associativity of matrix multiplication. The standard proof that I have in my text book is similar to the one found here. While I do understand the proof I would have a hard time recreating it, say in an exam. However, since each matrix corresponds to a linear transformation and matrix multiplication corresponds to composition of said transformations it seems like proving associativity of composition (Which is trivial) is a sufficient proof. thoughts?

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    I agree. That is: it suffices to note that $(AB)x = A(Bx)$ for all vectors $x$. – 2017-02-13
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    You are presupposing, however, that a matrix is completely determined by its associated linear map. – 2017-02-13
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    But you have to add sentences like "fix a basis" and so on.. – 2017-02-13
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    @Omnomnomnom how is a matrix not determined by its associated linear map? – 2017-02-13
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    @IttayWeiss it is, but I'm just saying that constitutes another layer to the proof (especially if we're starting from scratch, which textbooks often do when proving this). – 2017-02-13
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    Ah, yes. But it is trivial. – 2017-02-13
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    @IttayWeiss nothing is trivial when you're learning linear algebra for the first time. – 2017-02-13
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    Fair enough. But some things are more trivial than others ;) – 2017-02-13

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Absolutely yes, and this line of thought leads to extremely clear proofs. You have to just once prove a handful of properties to set up the relationship between matrices and linear transformations, and you're good to go. Do note though that the difficulty with establishing associativity becomes the proof that matrix multiplication indeed corresponds to composition. So some persistence of difficulty remains. I find it very unfortunate that so many books do not emphasize the linear transformations approach.