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Mathematics-Online course: Linear Algebra - Linear Systems of Equations - Direct Methods

Cramer's Rule

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For a square LSE $ Ax=b$ 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 $ a_j$ denotes the $ j$-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)

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  automatically generated 4/21/2005