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  "body": "\n## Statement\n\nLet $R$ be a ring.\nLet $A$ be an $m \\times n$ matrix over $R$, and let $B$ and $C$ be $n \\times p$ matrices over $R$.\nThen $A(B + C) = AB + AC$.\nLet $D$ and $E$ be $m \\times n$ matrices over $R$, and let $G$ be an $n \\times p$ matrix over $R$.\nThen $(D + E)G = DG + EG$.\n\n## Remarks\n\nBoth identities follow entry by entry from the distributive law in $R$.\nWe need both identities, because [[matrix multiplication|032Q9FB8MT]] is not commutative.\nTogether with [[associativity|073E4XC016]], they make the square matrices a [[ring|0FHVPRAE60]].\n\n## Explanations\n\n### L1 human\n\nMultiplication of matrices spreads over addition, as for numbers.\nFor example, $A(B + C)$ is the same as $AB + AC$.\nWe must keep the order: $A$ stays on the left in both terms.\n",
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