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  "body": "\n## Definition\n\nLet $R$ be a ring, let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $R$, and let $B = (b_{jk})$ be an $n \\times p$ matrix over $R$.\nThe *product* $AB$ is the $m \\times p$ matrix with entries\n$$(AB)_{ik} = \\sum_{j=1}^{n} a_{ij} b_{jk}$$\nfor $1 \\le i \\le m$ and $1 \\le k \\le p$.\n\n## Remarks\n\nThe product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$.\nColumn $k$ of $AB$ is the [[matrix–vector product|0PDVQEP0ZC]] of $A$ with column $k$ of $B$.\nThe product is [[associative|073E4XC016]] and [[distributive|0RXD6EZYBK]], but it is [[not commutative|0P506SPZP2]].\nIf $A$ and $B$ are the matrices of two linear maps, then $AB$ is the matrix of their composition.\n\n## Explanations\n\n### L1 human\n\nTo find the entry in row $i$ and column $k$ of $AB$, go along row $i$ of $A$ and down column $k$ of $B$.\nMultiply the pairs of numbers and add the products.\nFor example, $\\begin{pmatrix} 1 \u0026 2 \\\\ 0 \u0026 1 \\end{pmatrix} \\begin{pmatrix} 3 \\\\ 4 \\end{pmatrix} = \\begin{pmatrix} 11 \\\\ 4 \\end{pmatrix}$.\n",
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