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  "body": "\n## Statement\n\nLet $K$ be a commutative ring and let $A$ and $B$ be $n \\times n$ matrices over $K$.\nThen\n$$\\det(AB) = \\det A \\cdot \\det B,$$\nwhere $AB$ is the [[matrix product|032Q9FB8MT]] and $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nThus $\\det$ is a group homomorphism from the [[general linear group|0FCHE902D7]] $GL_n(F)$ to $F^\\times$, over a field $F$.\nIt follows that $\\det(AB) = \\det(BA)$, although $AB \\ne BA$ in general.\nSimilar matrices have the same determinant, because $\\det(P^{-1} A P) = \\det A$.\n\n## Explanations\n\n### L1 human\n\nIf one matrix scales volume by $\\det A$ and another by $\\det B$, then doing both scales volume by $\\det A \\cdot \\det B$.\nThis is the meaning of the product rule.\nIt is one of the most useful properties of the determinant.\n",
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