Cramer's rule

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, Cramer's rule is an explicit formula for the solution of a <a href="/facts/System_of_linear_equations/fk9CFdgp">system of linear equations</a> with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the <a href="/facts/Determinant/eGYqJ2TF">determinants</a> of the (square) <a href="/facts/Coefficient_matrix/S5JmDEW6">coefficient matrix</a> and of <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after <a href="/facts/Gabriel_Cramer/x10LLrVj">Gabriel Cramer</a>, who published the rule for an arbitrary number of unknowns in 1750, although <a href="/facts/Colin_Maclaurin/nVexkojA">Colin Maclaurin</a> also published special cases of the rule in 1748, and possibly knew of it as early as 1729.
Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while <a href="/facts/Gaussian_elimination/23Rvf4ht">Gaussian elimination</a> produces the result with the same (up to a constant factor independent of ⁠
 
 
 
 n
 
 
 {\displaystyle n}
 
⁠) <a href="/facts/Computational_complexity/Qenh22Ja">computational complexity</a> as the computation of a single determinant. Moreover, <a href="/facts/Bareiss_algorithm/uUKd5hKv">Bareiss algorithm</a> is a simple modification of Gaussian elimination that produces in a single computation a matrix whose nonzero entries are the determinants involved in Cramer's rule.

Cramer's rule open-in-new

Cramer's rule