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  "body": "\n## Definition\n\nLet $R$ be a ring, for example a field, and let $m, n \\ge 0$ be integers.\nAn $m \\times n$ *matrix* over $R$ is a family $A = (a_{ij})$ of elements $a_{ij} \\in R$, indexed by the pairs $(i, j)$ with $1 \\le i \\le m$ and $1 \\le j \\le n$.\nThe element $a_{ij}$ is the *entry* of $A$ in row $i$ and column $j$.\nThe notation for $A$ is a rectangular array with $m$ rows and $n$ columns:\n$$A = \\begin{pmatrix} a_{11} \u0026 \\cdots \u0026 a_{1n} \\\\ \\vdots \u0026 \\ddots \u0026 \\vdots \\\\ a_{m1} \u0026 \\cdots \u0026 a_{mn} \\end{pmatrix}.$$\nThe set of all $m \\times n$ matrices over $R$ is written $M_{m \\times n}(R)$ or $R^{m \\times n}$.\n\n## Remarks\n\nTwo matrices are equal when they have the same size and the same entries in each position.\nA $1 \\times n$ matrix is a *row vector*, and an $m \\times 1$ matrix is a *column vector*.\nMathlib uses arbitrary finite index types in place of $\\{1, \\dots, m\\}$ and $\\{1, \\dots, n\\}$.\n\n## Explanations\n\n### L1 human\n\nA matrix is a table of numbers with a fixed number of rows and columns.\nThe matrix $\\begin{pmatrix} 1 \u0026 2 \u0026 3 \\\\ 4 \u0026 5 \u0026 6 \\end{pmatrix}$ has 2 rows and 3 columns, so it is a $2 \\times 3$ matrix.\nWe name an entry by its row first and then its column.\n",
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