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Zwei einfache Aufgaben zur Matrizenrechnung.

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Matrizenrechnung

Berechnen Sie die unten angegebenen Matrizen.

Achten Sie bitte darauf, ggf. zunächst die korrekte Größe der Ergebnis-Matrix festzulegen!

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Lösung zu a)

Bei der transponierten Matrix werden einfach nur Zeilen und Spalten vertauscht, das Ergebnis ist demnach

$\\var{B}^T=\\var{BT}$.

Lösung zu b):

Allgemein gilt: Ist $A$ eine $(n,p)$-Matrix und $B$ eine $(p,m)$-Matrix dann ist die Produktmatrix $C = A \\cdot B$ ist eine $(m,n)$-Matrix, und die Elemente $c_{ik}$ ergeben sich als Skalarprodukt der $i$-ten Zeile von $A$ mit der $k$-ten Spalte von $B$, also:
$c_{ik} \\ = \\ \\sum\\limits_{r=1}^p a_{ir} b_{rk} \\qquad(i=1,\\ldots,m; \\ k=1,\\ldots,n).$

Im konkreten Fall ergibt dies:

$A$ ist eine $(3,3)$-Matrix und $B$ ist eine $(3,2)$-Matrix, also ist $C=A\\cdot B$ eine $(3,2)$-Matrix, es ergibt sich dann:

$\\var{A}\\cdot\\var{B}=\\begin{pmatrix}
\\var{A[0][0]}\\cdot \\var{B[0][0]}+ \\var{A[0][1]}\\cdot \\var{B[1][0]} + \\var{A[0][2]}\\cdot \\var{B[2][0]} &
\\var{A[0][0]}\\cdot \\var{B[0][1]}+ \\var{A[0][1]}\\cdot \\var{B[1][1]} + \\var{A[0][2]}\\cdot \\var{B[2][1]} \\\\
\\var{A[1][0]}\\cdot \\var{B[0][0]}+ \\var{A[1][1]}\\cdot \\var{B[1][0]} + \\var{A[1][2]}\\cdot \\var{B[2][0]} &
\\var{A[1][0]}\\cdot \\var{B[0][1]}+ \\var{A[1][1]}\\cdot \\var{B[1][1]} + \\var{A[1][2]}\\cdot \\var{B[2][1]} \\\\
\\var{A[2][0]}\\cdot \\var{B[0][0]}+ \\var{A[2][1]}\\cdot \\var{B[1][0]} + \\var{A[2][2]}\\cdot \\var{B[2][0]} &
\\var{A[2][0]}\\cdot \\var{B[0][1]}+ \\var{A[2][1]}\\cdot \\var{B[1][1]} + \\var{A[2][2]}\\cdot \\var{B[2][1]} \\\\
\\end{pmatrix}=\\var{AxB}$

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$\\var{B}^T=$ [[0]]

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$\\var{A}\\cdot\\var{B}=$

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