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  "body": "\n## Definition\n\nLet $\\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$ and let $a, b \\in S$.\nThe elements $a$ and $b$ *commute* if $a \\ast b = b \\ast a$.\n\n## Remarks\n\nExample: in the set of $2 \\times 2$ real matrices with matrix multiplication, every matrix $A$ commutes with $2A$, because $A(2A) = 2A^2 = (2A)A$.\n\nNon-example: the matrices\n$$A = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}, \\quad B = \\begin{pmatrix} 1 \u0026 0 \\\\ 1 \u0026 1 \\end{pmatrix}$$\ndo not commute, because $AB = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ while $BA = \\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$.\n\nThe relation \"commutes with\" is reflexive and symmetric but not transitive in general.\nIn the example, $A$ commutes with the identity matrix $I$ and $I$ commutes with $B$, but $A$ and $B$ do not commute.\nThe operation $\\ast$ is commutative if and only if every two elements of $S$ commute.\n",
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