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  "body": "\n## Statement\n\nLet $R$ be a commutative ring.\nLet $A$ be an $m \\times n$ matrix and $B$ an $n \\times p$ matrix over $R$.\nThen $(AB)^T = B^T A^T$, where $^T$ denotes the [[transpose|09DXWYRD43]].\n\n## Remarks\n\nThe order of the factors reverses.\nThe proof compares entries: both sides have entry $(k, i)$ equal to $\\sum_{j} a_{ij} b_{jk}$.\nBy induction, $(A_1 A_2 \\cdots A_k)^T = A_k^T \\cdots A_2^T A_1^T$.\nIf $R$ is not commutative, the identity can fail.\n\n## Explanations\n\n### L1 human\n\nThe transpose of a product is the product of the transposes in the reverse order.\nThis is like taking off socks and shoes: the last thing put on comes off first.\nThe sizes also force this order, because $B^T$ is $p \\times n$ and $A^T$ is $n \\times m$.\n",
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