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  "body": "\n## Statement\n\nLet $F$ be a field, let $A$ be an $m \\times n$ matrix and $B$ an $n \\times p$ matrix over $F$.\nThen\n$$\\rank(AB) \\le \\min(\\rank A, \\rank B),$$\nwhere $\\rank$ is the [[rank of a matrix|0ZC9YTXZRY]].\n\n## Remarks\n\nThe column space of $AB$ is contained in the column space of $A$, so $\\rank(AB) \\le \\rank A$.\nThe row space of $AB$ is contained in the row space of $B$, so $\\rank(AB) \\le \\rank B$.\nIf $A$ is invertible, then $\\rank(AB) = \\rank B$; if $B$ is invertible, then $\\rank(AB) = \\rank A$.\nA lower bound is Sylvester's rank inequality: $\\rank A + \\rank B - n \\le \\rank(AB)$.\n\n## Explanations\n\n### L1 human\n\nMultiplying by a matrix can never create new independent directions.\nSo the rank of a product is at most the rank of each factor.\nMultiplying by an invertible matrix keeps the rank.\n",
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