# Dictionary:Cramer’s rule

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(krā’ m∂rz) The solution to a set of ${\displaystyle n}$ linear equations in ${\displaystyle n}$ unknowns written compactly as

${\displaystyle Ax=b,}$

where

${\displaystyle a_{11}x_{1}+a_{12}x_{2}+...+a_{1n}x_{n}=b_{1}}$
${\displaystyle a_{21}x_{1}+a_{22}x_{2}+...+a_{2n}x_{n}=b_{2}}$
${\displaystyle a_{31}x_{1}+a_{32}x_{2}+...+a_{3n}x_{n}=b_{3}}$
${\displaystyle .\qquad \qquad .\qquad \qquad .\qquad \qquad .}$
${\displaystyle .\qquad \qquad .\qquad \qquad .\qquad \qquad .}$
${\displaystyle .\qquad \qquad .\qquad \qquad .\qquad \qquad .}$
${\displaystyle a_{n1}x_{1}+a_{n2}x_{2}+...+a_{nn}x_{n}=b_{n}}$

has solutions for each ${\displaystyle x_{i}}$ given by the ratio of determinants

${\displaystyle x_{i}={\frac {\det(A_{i})}{\det(A)}}}$

where the matrix ${\displaystyle A_{i}}$ is obtained by replacing the ${\displaystyle i}$-th column of the matrix ${\displaystyle A}$ with the vector b.

This is usually not the most economical way for computers to solve simultaneous equations. Named for Gabriel Cramer (1704–1752), French mathematician.