شاه کلید نکات
شاه کلید سوال
دترمینان ماتریس زیر را بیابید.
\(A = \left[ {\begin{array}{*{20}{c}}{\,\,\,\,\left| {\begin{array}{*{20}{c}}2&3\\1&2\end{array}} \right|}&{\,\left| {\begin{array}{*{20}{c}}4&2\\{ - 1}&1\end{array}\,\,\,} \right|}\\{ - \left| {\begin{array}{*{20}{c}}1&5\\2&1\end{array}} \right|}&{\left| {\begin{array}{*{20}{c}}2&{ - 3}\\1&5\end{array}} \right|}\end{array}} \right]\)
ابتدا تک تک آرایه های ماتریس را بدست می آوریم:
\(\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}{\,\,\,\,\left| {\begin{array}{*{20}{c}}2&3\\1&2\end{array}} \right|}&{\,\left| {\begin{array}{*{20}{c}}4&2\\{ - 1}&1\end{array}\,\,\,} \right|}\\{ - \left| {\begin{array}{*{20}{c}}1&5\\2&1\end{array}} \right|}&{\left| {\begin{array}{*{20}{c}}2&{ - 3}\\1&5\end{array}} \right|}\end{array}} \right]\\\\\\\left| {\begin{array}{*{20}{c}}2&3\\1&2\end{array}} \right| = 2 \times 2 - 3 \times 1 = 4 - 3 = 1\\\\\left| {\begin{array}{*{20}{c}}4&2\\{ - 1}&1\end{array}} \right| = 4 \times 1 - 2 \times ( - 1) = 6\\\\ - \left| {\begin{array}{*{20}{c}}1&5\\2&1\end{array}} \right| = - (1 \times 1 - 5 \times 2) = 9\\\\\left| {\begin{array}{*{20}{c}}2&{ - 3}\\1&5\end{array}} \right| = 2 \times 5 - ( - 3) \times 1 = 13\\\\ \Rightarrow A = \left[ {\begin{array}{*{20}{c}}1&6\\9&{13}\end{array}} \right]\end{array}\)
حال دترمینان ماتریس را بدست می آوریم:
\(\begin{array}{l}\left| A \right| = \left| {\begin{array}{*{20}{c}}1&6\\9&{13}\end{array}} \right| = 1 \times 13 - 6 \times 9 = \\\\13 - 54 = - 41\end{array}\)
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