> [!NOTE] Lemma > [[Real Matrix Product|Multiplication]] is [[Associativity|associative]]: that is, for all [[Real Matrices|real matrices]] $A,B,C$ with order $m\times n,$ $n\times l,$ and $l\times p$ respectively, we have $A(BC)=(AB)C.$ **Proof**: For all $(i,j)\in[1,2,\dots,m]\times[1,2,\dots,l],$ the $(i,j)$ entry of $A(BC)$ is given by $\sum_{r=1}^{n} a_{ir} \left( \sum_{s=1}^{l} b_{rs}c_{sj} \right) $while the $(i,j)$ entry of $(AB)C$ is given by $\sum_{s=1}^{l} \left( \sum_{r=1}^{n} a_{ir} b_{r_{s}} \right) c_{sj} $which are equal since the order of finite sums may be swapped without changing the result.