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  "body": "\n## Definition\n\nLet $R$ be a ring, let $n \\ge 0$ be an integer, and let $c \\in R$.\nThe *scalar matrix* of order $n$ with value $c$ is the [[diagonal matrix|044JNC7ZAA]] $\\operatorname{diag}(c, \\dots, c) = c I_n$, where $I_n$ is the [[identity matrix|06DW9BAWB1]].\nA square matrix is a *scalar matrix* when it equals $c I_n$ for some $c \\in R$.\n\n## Remarks\n\nOver a commutative ring, a scalar matrix commutes with every square matrix of the same order.\nOver a field, the scalar matrices are exactly the $n \\times n$ matrices that commute with every $n \\times n$ matrix.\nAs a linear map, $c I_n$ multiplies each vector by $c$.\n\n## Explanations\n\n### L1 human\n\nA scalar matrix has the same number at every place on the diagonal and zeros elsewhere.\nMultiplying by it is the same as multiplying by that number.\nFor example, $3 I_2 = \\begin{pmatrix} 3 \u0026 0 \\\\ 0 \u0026 3 \\end{pmatrix}$ triples each vector.\n",
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