0607_blatt05 - Proseminar zur Vorlesung Mathematik fr...

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Proseminar zur Vorlesung Mathematik für Wirtschaftswissenschaftler Wintersemester 2006/2007 Blatt 5 20.11. - 24.11.2006 Aufgabe 4.29: Für welche a, b R ist 2 - 1 - 1 a 1 / 4 b 1 / 8 1 / 8 - 1 / 8 = 1 2 4 0 1 6 1 3 2 - 1 ? Aufgabe 4.31: Gegeben sei die Matrix A = a b c d mit a, b, c, d R \ { 0 } (a) Zeigen Sie, dass A genau dann invertierbar ist, wenn ad - bc = 0 ist! (b) Invertieren Sie die Matrix A unter dieser Voraussetzung! Aufgabe 4.28: (a) Sind folgende Matrizen invertierbar? Berechnen Sie ggf. die Inverse. A = 1 0 2 3 1 2 - 2 4 0 , C = 1 2 3 4 - 5 8 1 7 0 2 - 2 3 0 0 0 0 , D = 1 1 1 1 1 - 5 0 0 0 0 1 - 1 0 2 - 1 0 (b) Berechnen Sie die Lösung für das Gleichungssystem Ax = b mit b = (1 , 1 , 1) T . Aufgabe 4.36: Gegeben seien die Matrizen A = 1 1 2 3 ; B = 1 2 0 0 1 1 2 1 2 (a) Bestimmen Sie det( A ) und det( B ) durch Anwenden der Vorlesungs-Formeln. (b) Bestimmen Sie nochmals det( B ) rekursiv, indem Sie geeignete Zeilen– bzw. Spal- tenentwicklungen benutzen, sowie det( a ) = a für a R .
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