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  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ and $B$ be $m \\times n$ matrices over $F$.\nThe matrix $A$ is *row equivalent* to $B$ when there is a finite sequence of [[elementary row operations|04XWH28452]] that changes $A$ into $B$.\n\n## Remarks\n\nRow equivalence is an equivalence relation, because each elementary row operation has an inverse operation.\nThe matrices $A$ and $B$ are row equivalent if and only if $B = PA$ for an [[invertible|0EB9N0S33C]] $m \\times m$ matrix $P$.\nEach matrix is row equivalent to exactly one matrix in [[reduced row echelon form|0WF5AQYTW6]].\nRow equivalent matrices have the same [[row space|069MCGX635]] and the same [[null space|0PHDYNR24D]].\n\n## Explanations\n\n### L1 human\n\nTwo matrices are row equivalent when row operations can turn one into the other.\nIf they are augmented matrices of two systems, then the two systems have the same solutions.\nRow reduction replaces a matrix by a simpler row equivalent matrix.\n",
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