Cramer sin reegel

Tekst üüb Öömrang

Cramer sin reegel as en metood, am linear liknangsüsteemen tu liasin. Hat as näämd efter Gabriel Cramer.

Bispal

(1) Det liknangsüsteem uun normoolfurem

${\begin{matrix}\color {blue}{1}\,\color {black}x_{1}+\color {blue}{2}\,\color {black}x_{2}=\color {green}{3}\\\color {blue}{4}\,\color {black}x_{1}+\color {blue}{5}\,\color {black}x_{2}=\color {green}{6}\end{matrix}}$

(2) Det ütjwidjet koefisientenmaatriks

${\begin{pmatrix}\color {blue}{A}&\color {green}b\end{pmatrix}}=\left({\begin{array}{cc|c}\color {blue}{1}&\color {blue}{2}&\color {green}{3}\\\color {blue}{4}&\color {blue}{5}&\color {green}{6}\end{array}}\right)$

(3) Liasang efter Cramer sin reegel (det=determinant)

$x_{1}={\frac {\det(A_{1})}{\det(A)}}={\frac {\det {\begin{pmatrix}\color {green}{3}&\color {blue}{2}\\\color {green}{6}&\color {blue}{5}\end{pmatrix}}}{\det {\begin{pmatrix}\color {blue}{1}&\color {blue}{2}\\\color {blue}{4}&\color {blue}{5}\end{pmatrix}}}}={\frac {3\cdot 5-6\cdot 2}{1\cdot 5-4\cdot 2}}={\frac {3}{-3}}=-1\qquad$

$x_{2}={\frac {\det(A_{2})}{\det(A)}}={\frac {\det {\begin{pmatrix}\color {blue}{1}&\color {green}{3}\\\color {blue}{4}&\color {green}{6}\end{pmatrix}}}{\det {\begin{pmatrix}\color {blue}{1}&\color {blue}{2}\\\color {blue}{4}&\color {blue}{5}\end{pmatrix}}}}={\frac {1\cdot 6-4\cdot 3}{1\cdot 5-4\cdot 2}}={\frac {-6}{-3}}=2$

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• Lineaar algebra