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Cramer's ruleInlinear algebra,Cramer's ruleis an explicit formula for the solution of asystem of linear equations with as many equations asunknowns, valid whenever the system has a unique solution. It expresses the solution in terms of thedeterminants of the (square)coefficientmatrix and of matrices obtained from it by replacing one column by the vector of right-hand-sides of the equations. It isnamed afterGabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750,[1][2]althoughColinMaclaurin also published special cases of the rule in 1748[3](and possibly knew of it as early as 1729).[4][5][6]Cramer's rule is computationally very inefficient for systems of more than two or three equations;[7]its asymptotic complexity isO((n+1)!) compared to elimination methods that have polynomial time complexity.[8][9]Cramer's rule is alsonumerically unstableeven for 2×2 systems.[10]General caseProofFinding inverse matrixApplicationsExplicit formulas for small systemsDifferential geometryRicci calculusComputing derivatives implicitlyInteger programmingOrdinary differential equationsGeometric interpretationOther proofsA proof by abstract linear algebraA short proofProof using Cliford algebraIncompatible and indeterminate casesReferencesExternal linksConsider a system ofnlinear equations fornunknowns, represented in matrix multiplication form as follows:where then×nmatrixAhas a nonzero determinant, and the vectoris the column vector of the variables. Thenthe theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by:whereis the matrix formed by replacing thei-th column ofAby the column vectorb.ContentsGeneral case
A more general version of Cramer's rule[11]considers the matrix equationwhere then×nmatrixAhas a nonzero determinant, andX,Baren×mmatrices. Given sequencesand, letbe thek×ksubmatrix ofXwith rows inand columns in. Letbe then×nmatrix formed by replacing thecolumn ofAby thecolumn ofB, for all. ThenIn the case, this reduces to the normal Cramer's rule.The rule holds for systems of equations with coefficients and unknowns in anyfield, not just in thereal numbers. It has recently beenshown that Cramer's rule can be implemented in O(n3) time,[12]which is comparable to more common methods of solving systems oflinear equations, such asGaussian elimination (consistently requiring 2.5 times as many arithmetic operations for all matrix sizes,while exhibiting comparable numeric stability in most cases).The proof for Cramer's rule uses just two properties of determinants: linearity with respect to any given column (taking for thatcolumn alinear combination of column vectors produces as determinant the corresponding linear combination of their determinants),and the fact that the determinant is zero whenever two columns are equal (which is implied by the basic property that the sign of thedeterminant flips if you switch two columns).Fix the indexj

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