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  "body": "\n## Statement\n\nLet $F$ be a field, let $A$ be an $n \\times n$ matrix over $F$ with $\\det A \\ne 0$, and let $b \\in F^n$.\nFor $1 \\le i \\le n$, let $A_i$ be the matrix that results when column $i$ of $A$ is replaced by $b$.\nThen the [[linear system|0GT4WWGWH2]] $Ax = b$ has exactly one solution $x = (x_1, \\dots, x_n)$, and\n$$x_i = \\frac{\\det A_i}{\\det A} \\qquad (1 \\le i \\le n).$$\n\n## Remarks\n\nFor $n = 2$, the system $ax + by = e$, $cx + dy = f$ has the solution $x = (ed - bf)/(ad - bc)$ and $y = (af - ec)/(ad - bc)$.\nThe rule follows from the formula for the inverse with the [[adjugate|07W9H8QF38]].\nIt needs $n + 1$ determinants, so it is slow for large $n$ compared with [[Gaussian elimination|0JVJ2WCXZH]].\n\n## Explanations\n\n### L1 human\n\nCramer's rule gives each unknown as a quotient of two determinants.\nThe denominator is always $\\det A$.\nFor the numerator of unknown $i$, put the right side $b$ in column $i$ of $A$.\n",
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