Metoda Gauss - Jordan

axa Dum Noi 06, 2011 12:12 am

Se da sistemul :

$2x-y-2z+3t=\alpha$
$x-2y+z-t=\beta$
$3x+y-z+at=\gamma$
$x-3y+bz+4t=\delta$
$4x-by+cz+t=\varepsilon$

unde a=2 , b=-3 , c=0 .

Nu stiu sa-l finalizez , dar e posibil sa gresesc si pe parcurs .
Puteti sa`mi aratati o rezolvare completa ? Trebuie sa folesesc metoda Gauss-Jordan sau Gauss.

Multumesc .

**Admin** Lun Noi 07, 2011 11:49 pm

Sper ca nu am greșit la calcule :
$\left( {\begin{array}{*{20}{c}} 2&{ - 1}&{ - 2}&3&\alpha \\ 1&{ - 2}&1&{ - 1}&\beta \\ 3&1&{ - 1}&2&\gamma \\ 4&3&0&1&\varepsilon \end{array}} \right) \to \left( {\begin{array}{*{20}{c}} 0&3&{ - 4}&5&{\alpha - 2\beta }\\ 1&{ - 2}&1&{ - 1}&\beta \\ 0&7&{ - 4}&5&{\gamma - 3\beta }\\ 0&{11}&{ - 4}&5&{\varepsilon - 4\beta } \end{array}} \right) \to \left( {\begin{array}{*{20}{c}} 0&0&{ - 4}&5&{\frac{7}{4}\alpha - \frac{5}{4}\beta - \frac{3}{4}\gamma }\\ 1&0&1&{ - 1}&{ - \frac{1}{2}a + \frac{1}{2}\beta + \frac{1}{2}\gamma }\\ 0&1&0&0&{ - \frac{1}{4}\alpha - \frac{1}{4}\beta + \frac{1}{4}\gamma }\\ 0&0&0&0&{\frac{1}{2}\varepsilon - \gamma + \frac{1}{2}\alpha } \end{array}} \right)$
$\to \left( {\begin{array}{*{20}{c}} 0&0&1&{ - \frac{5}{4}}&{ - \frac{7}{{16}}\alpha + \frac{5}{{16}}\beta + \frac{3}{{16}}\gamma }\\ 1&0&0&{\frac{1}{4}}&{ - \frac{1}{{16}}a + \frac{3}{{16}}\beta + \frac{5}{{16}}\gamma }\\ 0&1&0&0&{ - \frac{1}{4}\alpha - \frac{1}{4}\beta + \frac{1}{4}\gamma }\\ 0&0&0&0&{\varepsilon - 2\gamma + \alpha } \end{array}} \right)$

Pentru simplitate voi nota : $\varepsilon = 2\gamma - \alpha ,\;t \in {\Cal R}$
Atunci vom obține:
$\begin{array}{l} x = - {\textstyle{1 \over 4}}t - {\textstyle{1 \over {16}}}a + {\textstyle{3 \over {16}}}\beta + {\textstyle{5 \over {16}}}\gamma \\ y = - {\textstyle{1 \over 4}}\alpha - {\textstyle{1 \over 4}}\beta + {\textstyle{1 \over 4}}\gamma \\ z = {\textstyle{5 \over 4}}t - {\textstyle{7 \over {16}}}\alpha + {\textstyle{5 \over {16}}}\beta + {\textstyle{3 \over {16}}}\gamma \; \end{array}$

**Reclama** Dum Mai 13, 2012 4:10 am

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Metoda Gauss - Jordan

Metoda Gauss - Jordan

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