Mathematics-Online lexicon: Cramer's Rule

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	Mathematics-Online lexicon:
	Cramer's Rule

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For a square LSE

the following holds true:

$\displaystyle x_i \operatorname{det} A = \operatorname{det} (a_1,\ldots,a_{i-1},b,a_{i+1},\ldots,a_n)\, ,$

where

denotes the

-th column of the coefficient matrix.

In particular, if $\operatorname{det}A\ne0$ , then we can find the inverse $B=A^{-1}$ by

$\displaystyle b_{i,j} = \frac{ \operatorname{det} (a_1,\ldots,a_{i-1},e_j,a_{i+1},\ldots,a_n)}{ \operatorname{det}A}\, .$

(Authors: Burkhardt/Höllig/Streit)

Annotation:

automatically generated 3/16/2005