TU Wien:Mathematik 1 UE (diverse)/Übungen WS06/Beispiel 559

Angabe

Man berechne

$\left|\begin{array}{cccc}2& 4& -1& 3\\ 1& 2& 0& -1\\ 1& 2& 7& 4\\ 4& 5& 6& 6\end{array}\right|$

Lösungsvorschlag von mnemetz

$\begin{array}{c}\left|\begin{array}{cccc}2& 4& -1& 3\\ 1& 2& 0& -1\\ 1& 2& 7& 4\\ 4& 5& 6& 6\end{array}\right|=(-1)\ast (-1{)}^{3+1}\ast \left|\begin{array}{ccc}1& 2& -1\\ 1& 2& 4\\ 4& 5& 6\end{array}\right|+0+7\ast (-1{)}^{3+3}\ast \left|\begin{array}{ccc}2& 4& 3\\ 1& 2& -1\\ 4& 5& 6\end{array}\right|+6\ast (-1{)}^{3+4}\ast \left|\begin{array}{ccc}2& 4& -3\\ 1& 2& -1\\ 1& 2& 4\end{array}\right|=\\ =-(12+32-5+8-20-12)+7\ast (24-16+15-24+10-24)-6\ast (16-4+6-6+4-16)=\\ =-15-7\ast 15=-8\ast 15=\mathbf{-}\mathrm{𝟏𝟐𝟎}\end{array}$

Angabe SS07

F?ür die Matrizen A, B mit

$A=\left(\begin{array}{ccc}1& 3& 2\\ 2& 4& 6\\ -1& -2& 2\end{array}\right)$ $B=\left(\begin{array}{ccc}-1& 3& 2\\ 2& -4& 6\\ 1& -2& 2\end{array}\right)$

bestimme man C = AB und verifiziere den Determinantensatz detC = detA *detB

Lösungsvorschlag von Jacko

Nützliches

$C=AB=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)\ast \left(\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right)$ =

$\left(\begin{array}{ccc}{a}_{11}{b}_{11}+a_{12}{b}_{21}+a_{13}{b}_{31}& a_{11}{b}_{12}+a_{12}{b}_{22}+a_{13}{b}_{32}& a_{11}{b}_{13}+a_{12}{b}_{23}+a_{13}{b}_{33}\\ a_{21}{b}_{11}+a_{22}{b}_{21}+a_{23}{b}_{31}& a_{21}{b}_{12}+a_{22}{b}_{22}+a_{23}{b}_{32}& a_{21}{b}_{13}+a_{22}{b}_{23}+a_{23}{b}_{33}\\ a_{31}{b}_{11}+a_{32}{b}_{21}+a_{33}{b}_{31}& a_{31}{b}_{12}+a_{32}{b}_{22}+a_{33}{b}_{32}& a_{31}{b}_{13}+a_{32}{b}_{23}+a_{33}{b}_{33}\end{array}\right)$

Lösung für Multiplikation

$C=AB=\left(\begin{array}{ccc}1& 3& 2\\ 2& 4& 6\\ -1& -2& 2\end{array}\right)\ast \left(\begin{array}{ccc}-1& 3& 2\\ 2& -4& 6\\ 1& -2& 2\end{array}\right)=\left(\begin{array}{ccc}7& -13& 24\\ 12& -22& 40\\ -1& 1& -10\end{array}\right)$

$D=BA=\left(\begin{array}{ccc}-1& 3& 2\\ 2& -4& 6\\ 1& -2& 2\end{array}\right)\ast \left(\begin{array}{ccc}1& 3& 2\\ 2& 4& 6\\ -1& -2& 2\end{array}\right)=\left(\begin{array}{ccc}3& 5& 20\\ -12& -22& -8\\ -5& -9& -6\end{array}\right)$

Links

Tool zum Überprüfen von Matrizenmultiplikation