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  "body": "\n## Definition\n\nLet $F$ be a field and let $n \\ge 1$ be an integer.\nAn $n \\times n$ *elementary matrix* over $F$ is a matrix that results from the [[identity matrix|06DW9BAWB1]] $I_n$ by one [[elementary row operation|04XWH28452]].\nThere are three types:\n\n1. $P_{ij}$, for $i \\ne j$: exchange rows $i$ and $j$ of $I_n$.\n2. $D_i(c)$, for $c \\ne 0$: multiply row $i$ of $I_n$ by $c$.\n3. $T_{ij}(c)$, for $i \\ne j$ and $c \\in F$: add $c$ times row $j$ of $I_n$ to row $i$.\n\n## Remarks\n\nFor each $n \\times p$ matrix $A$, the product $EA$ is the result of the same row operation on $A$.\nThe product $AE$ performs the matching column operation on $A$.\nEach elementary matrix is invertible, and its inverse is an elementary matrix of the same type.\nEach invertible matrix over a field is a product of elementary matrices.\n\n## Explanations\n\n### L1 human\n\nAn elementary matrix stores one row operation as a matrix.\nTo do the operation on $A$, we multiply $A$ on the left by this matrix.\nThen a whole elimination becomes a product of elementary matrices.\n",
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