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  "body": "\n## Summary\n\nTwo square matrices $A$ and $B$ commute when $AB = BA$.\n[[Matrix multiplication|032Q9FB8MT]] is [[not commutative|0P506SPZP2]] in general, so commuting pairs are special.\nCommuting matrices behave much like numbers: for example, $e^{A + B} = e^{A} e^{B}$ for [[commuting matrices|04BD23PSK5]].\nDiagonalisable matrices can be diagonalised by one common basis exactly when they commute, by [[simultaneous diagonalisation|0FK17N4DKA]].\n\n## Explanations\n\n### L1 human\n\nFor numbers, the order of a product never matters, but for matrices it usually does.\nMatrices that commute are the pairs for which the order does not matter.\nAs maps, applying $A$ then $B$ gives the same result as applying $B$ then $A$.\nFor example, every matrix commutes with itself, with its powers, and with every [[scalar matrix|067NKSBEWT]] $cI$.\nThe matrices $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \u0026 0 \\\\ 1 \u0026 1 \\end{pmatrix}$ do not commute.\n",
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