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  "body": "\n## Definition\n\nLet $R$ be a ring, let $c \\in R$, and let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $R$.\nThe *scalar multiple* $cA$ is the $m \\times n$ matrix with entries\n$$(cA)_{ij} = c\\, a_{ij}$$\nfor $1 \\le i \\le m$ and $1 \\le j \\le n$.\n\n## Remarks\n\nOver a field $F$, matrix addition and scalar multiplication make $M_{m \\times n}(F)$ a [[vector space|000AYPA9AJ]] of dimension $mn$.\nThe rules $c(A + B) = cA + cB$, $(c + d)A = cA + dA$, $(cd)A = c(dA)$ and $1A = A$ hold.\nFor example, $2 \\begin{pmatrix} 1 \u0026 -1 \\\\ 0 \u0026 3 \\end{pmatrix} = \\begin{pmatrix} 2 \u0026 -2 \\\\ 0 \u0026 6 \\end{pmatrix}$.\n\n## Explanations\n\n### L1 human\n\nMultiplying a matrix by a number means multiplying every entry by that number.\nThe size of the matrix does not change.\nDoubling a matrix doubles every number in the table.\n",
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