##\begin{align}[A(BC)]_{ij} &= \sum_r A_{ir}(BC)_{rj} \\ &= \sum_r A_{ir} \sum_s B_{rs}C_{sj}\\ &= \sum_r\sum_s A_{ir}B_{rs}C_{sj}\\ &= \sum_{s} (\sum_{r} A_{ir} B_{rs}) C_{sj} \\ &= [(AB) C]_{ij}\end{align}##(adsbygoogle = window.adsbygoogle || []).push({});

How did it went from ##2## to ##3##. In general is there a proof that sums can be switched like this ?

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# B Associativity of Matrix multiplication

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