Distributivity of matrix multiplication

Statement

Let $R$ be a ring. Let $A$ be an $m \times n$ matrix over $R$, and let $B$ and $C$ be $n \times p$ matrices over $R$. Then $A(B + C) = AB + AC$. Let $D$ and $E$ be $m \times n$ matrices over $R$, and let $G$ be an $n \times p$ matrix over $R$. Then $(D + E)G = DG + EG$.

Remarks

Both identities follow entry by entry from the distributive law in $R$. We need both identities, because matrix multiplication is not commutative. Together with associativity, they make the square matrices a ring.

Explanations

L1 human

Multiplication of matrices spreads over addition, as for numbers. For example, $A(B + C)$ is the same as $AB + AC$. We must keep the order: $A$ stays on the left in both terms.