> [!NOTE] Lemma > [[Real Matrix Product|Multiplication]] is [[Distributivity|distributive]] over [[Real Matrix Addition|addition of real matrices]]: that is, whenever the following products and sums make sense, we have $A(B+C)=AB+BC$and $(A+B)C=AC+BC.$ **Proof**: Suppose $A$ has order $m\times n$ and $B,C$ have order $n\times l.$ For all $(i,j)\in[1,2,\dots,m]\times[1,2,\dots,l],$ the $(i,j)$ entry of $A(B+C)$ is given by $\sum_{r=1}^{n} a_{ir}(b_{rj}+c_{rj})=\sum_{r=1}^{n} a_{ir}b_{rj} + \sum_{r=1}^{n} a_{ir}c_{rj}$which equals the $(i,j)$ entry of $AB+BC.$ Also note that $A(B+C)$ and $AB+BC$ both have order $m\times l$ thus $A(B+C)=AB+BC.$ Similarly $(A+B)C=AC+BC.$