Man berechne
|24−13120−112744566|=(−1)∗(−1)3+1∗|12−1124456|+0+7∗(−1)3+3∗|24312−1456|+6∗(−1)3+4∗|24−312−1124|==−(12+32−5+8−20−12)+7∗(24−16+15−24+10−24)−6∗(16−4+6−6+4−16)==−15−7∗15=−8∗15=−120{\displaystyle {\begin{aligned}{\begin{vmatrix}2&4&-1&3\\1&2&0&-1\\1&2&7&4\\4&5&6&6\end{vmatrix}}=(-1)*(-1)^{3+1}*{\begin{vmatrix}1&2&-1\\1&2&4\\4&5&6\end{vmatrix}}+0+7*(-1)^{3+3}*{\begin{vmatrix}2&4&3\\1&2&-1\\4&5&6\end{vmatrix}}+6*(-1)^{3+4}*{\begin{vmatrix}2&4&-3\\1&2&-1\\1&2&4\end{vmatrix}}=\\=-(12+32-5+8-20-12)+7*(24-16+15-24+10-24)-6*(16-4+6-6+4-16)=\\=-15-7*15=-8*15=\mathbf {-120} \end{aligned}}}
F?ür die Matrizen A, B mit
A=(132246−1−22){\displaystyle A={\begin{pmatrix}1&3&2\\2&4&6\\-1&-2&2\end{pmatrix}}} B=(−1322−461−22){\displaystyle B={\begin{pmatrix}-1&3&2\\2&-4&6\\1&-2&2\end{pmatrix}}}
bestimme man C = AB und verifiziere den Determinantensatz detC = detA *detB
C=AB=(a11a12a13a21a22a23a31a32a33)∗(b11b12b13b21b22b23b31b32b33){\displaystyle C=AB={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}*{\begin{pmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{pmatrix}}} =
(a11b11+a12b21+a13b31a11b12+a12b22+a13b32a11b13+a12b23+a13b33a21b11+a22b21+a23b31a21b12+a22b22+a23b32a21b13+a22b23+a23b33a31b11+a32b21+a33b31a31b12+a32b22+a33b32a31b13+a32b23+a33b33){\displaystyle {\begin{pmatrix}a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31}&a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32}&a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33}\\a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31}&a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32}&a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}\\a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31}&a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32}&a_{31}b_{13}+a_{32}b_{23}+a_{33}b_{33}\end{pmatrix}}}
C=AB=(132246−1−22)∗(−1322−461−22)=(7−132412−2240−11−10){\displaystyle C=AB={\begin{pmatrix}1&3&2\\2&4&6\\-1&-2&2\end{pmatrix}}*{\begin{pmatrix}-1&3&2\\2&-4&6\\1&-2&2\end{pmatrix}}={\begin{pmatrix}7&-13&24\\12&-22&40\\-1&1&-10\end{pmatrix}}}
D=BA=(−1322−461−22)∗(132246−1−22)=(3520−12−22−8−5−9−6){\displaystyle D=BA={\begin{pmatrix}-1&3&2\\2&-4&6\\1&-2&2\end{pmatrix}}*{\begin{pmatrix}1&3&2\\2&4&6\\-1&-2&2\end{pmatrix}}={\begin{pmatrix}3&5&20\\-12&-22&-8\\-5&-9&-6\end{pmatrix}}}
Tool zum Überprüfen von Matrizenmultiplikation
=
