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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cramer's rule ： ウィキペディア英語版 Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748 (and possibly knew of it as early as 1729).〔 〕〔 〕〔 〕 Cramer's rule is computationally very inefficient for systems of more than two or three equations; its asymptotic complexity is O(n·n!) compared to elimination methods that have polynomial time complexity. Cramer's rule is also numerically unstable even for 2×2 systems. ==General case== Consider a system of linear equations for unknowns, represented in matrix multiplication form as follows: :$Ax\; =\; b$ where the matrix has a nonzero determinant, and the vector $x\; =\; (x\_1,\; \backslash ldots,\; x\_n)^\backslash mathrm$ is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by: :$x\_i\; =\; \backslash frac\; \backslash qquad\; i\; =\; 1,\; \backslash ldots,\; n$ where $A\_i$ is the matrix formed by replacing the -th column of by the column vector . A more general version of Cramer's rule considers the matrix equation :$AX\; =\; B$ where the matrix has a nonzero determinant, and , are matrices. Given sequences and , let $X\_$ be the submatrix of with rows in $I\; :=\; (i\_1,\; \backslash ldots,\; i\_k\; )$ and columns in $J\; :=\; (j\_1,\; \backslash ldots,\; j\_k\; )$. Let $A\_(I,J)$ be the matrix formed by replacing the $i\_s$ column of by $j\_s$ column of B, for all $s\; =\; 1,\backslash ldots,\; k$. Then :$\backslash det\; X\_\; =\; \backslash frac.$ In the case $k\; =\; 1$, this reduces to the normal Cramer's rule. The rule holds for systems of equations with coefficients and unknowns in any field, not just in the real numbers. It has recently been shown that Cramer's rule can be implemented in O(''n''3) time, which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination (consistently requiring 2.5 times as many arithmetic operations for all matrix sizes, while exhibiting comparable numeric stability in most cases). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cramer's rule」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.