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WiMa-B-02a Wirtschaftsmathematik 2/WS1718 WiMa-B-02a.tex
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429
WiMa-B-02a Wirtschaftsmathematik 2/WS1718 WiMa-B-02a.tex
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\input{../settings/settings}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\begin{document}
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\klausur{WiMa-B-02a Wirtschaftsmathematik 2}
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{Prof. Dr. Christian Aßmann}
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{Wintersemester 17/18}
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{60}
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{Taschenrechner, Formelsammlung, ein handbeschriebenes DIN A4 Blatt (beidseitig)}
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\section*{Aufgabe 1 (4 Punkte)}
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Bestimmen Sie, falls möglich, den Homogenitätsgrad der Funktion
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\begin{equation*}
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f: \mathbb{R}^{+}\times \mathbb{R}^{+} \to \mathbb{R}, \qquad f(x,y) = y \ln(2^{x})
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\end{equation*}
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\begin{enumerate}[label=\Alph*)]
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\item 0
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\item 1
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\item 2
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\item $\ln(2)$
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\item $f(x,y)$ ist nicht homogen
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\end{enumerate}
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\section*{Aufgabe 2 (4 Punkte)}
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Die Funktion
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\begin{equation*}
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f(x) = \frac{x+2}{x-1},
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\end{equation*}
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\noindent
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mit $x\neq 1$, besitzt an der Stelle $x_{0}=2$ die Tangente
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\begin{equation*}
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g(x) = -3x+10,
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\end{equation*}
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\noindent
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was sie nicht zu prüfen brauchen.
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Approximieren Sie den Funktionswert von $f(x)$ an der Stelle $x_{1}=3$ durch die Tangende
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und geben Sie den absoluten Approximationsfehler $\delta f$ an.
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\begin{enumerate}[label=\Alph*)]
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\item $\delta f = 3$
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\item $\delta f = 0$
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\item $\delta f = 1,5$
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\item $\delta f = 3,5$
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\item $\delta f= -1,5$
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\end{enumerate}
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\newpage
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\section*{Aufgabe 3 (6 Punkte)}
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Bestimmen Sie für die Funktion
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\begin{equation*}
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f: = \mathbb{R} \to \mathbb{R}, \qquad f(x)=\sqrt{x^{2}+1}
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\end{equation*}
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\noindent
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das Taylorpolynom 2. Grades $T^{2}_{f}(x)$ mit der Entwicklungsstelle $x_{0}=0$.
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\begin{enumerate}[label=\Alph*)]
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\item $1+\frac{1}{2}x^{2}$
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\item $1+\frac{1}{2}x-\frac{1}{4}x^{2}$
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\item $x+\frac{1}{2}x^{2}$
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\item $1-\frac{1}{4}x^{2}$
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\item $\frac{1}{2}x^{2}$
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\end{enumerate}
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\section*{Aufgabe 4 (5 Punkte)}
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Welche der folgenden Aussagen ist falsch?
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\begin{enumerate}[label=\Alph*)]
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\item Das Matrizenprodukt $A \cdot B$ ist definiert für zwei beliebige Matrizen
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$A \in \mathcal{M}_{m,n}$ und $B \in \mathcal{M}_{r,s}$ mit $n=r$.
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\item Der Nullvektor ist immer linear unabhängig.
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\item Ist eine Matrix $\tilde{A}$ durch endlich viele elementare Zeilenumformungen aus
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der Matrix $A$ entstanden, so besitzen $A$ und $\tilde{A}$ den gleichen Rang.
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\item Eine Matrix $A \in \mathcal{M}_{n}$ ist invertierbar, wenn $rg(A)=n$ gilt.
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\item Die Lösungsmenge eines homogenen linearen Gleichungssystems enthält immer den Nullpunkt.
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\end{enumerate}
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\section*{Aufgabe 5 (6 Punkte)}
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Lösen Sie das bestimmte Integral
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\begin{equation*}
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\int^{b}_{1} x\ln(x)dx,
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\end{equation*}
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\noindent
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mit $b>1$, mittels partieller Integration.
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\begin{enumerate}[label=\Alph*)]
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\item $b\ln(b)-b+1$
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\item $\frac{1}{2}b^{2}\ln(b)-\frac{1}{4}b^{2}+\frac{1}{4}$
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\item $b-1$
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\item $b\ln(b)$
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\item $\frac{b^{2}-1}{2}\ln(b)$
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\end{enumerate}
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\section*{Aufgabe 6 (4 Punkte)}
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Berechnen Sie
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\begin{equation*}
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(a+b)^{T}\cdot c
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\end{equation*}
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\noindent
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für die Vektoren
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\begin{equation*}
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a=
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\begin{pmatrix}
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1\\
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4\\
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-2\\
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\end{pmatrix}
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, \qquad
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b=
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\begin{pmatrix}
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0\\
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-3\\
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3\\
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\end{pmatrix}
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, \qquad
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c=
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\begin{pmatrix}
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8\\
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5\\
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2\\
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\end{pmatrix}
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.
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\end{equation*}
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\begin{enumerate}[label=\Alph*)]
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\item 13.
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\item 14.
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\item 15.
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\item 16.
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\item 17.
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\end{enumerate}
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\section*{Aufgabe 7 (5 Punkte)}
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Seien die Matrizen
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\begin{equation*}
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A \in \mathcal{M}_{m,n}, B \in \mathcal{M}_{p,q} und C \in \mathcal{M}_{r,s}
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\end{equation*}
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\noindent
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Welche Bedingungen müssen $m,n,p,q,r,s \in \mathbb{N}$ erfüllen, damit
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\begin{equation*}
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A \cdot (B+C^{T})
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\end{equation*}
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\noindent
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definiert ist?
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\begin{enumerate}[label=\Alph*)]
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\item $n=p=r$
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\item $m=r$ und $p=q=s$
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\item $n=p=s$ und $q=r$
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\item $m=s$ und $p=q=r$
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\item $m=p=s$ und $n=q=r$
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\end{enumerate}
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\newpage
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\section*{Aufgabe 8 (5 Punkte)}
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Berechnen Sie die fehlenden Werte $\theta _{1}$ und $\theta _{2}$ in der Matrizenmultiplikation
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\begin{equation*}
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\begin{pmatrix}
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\theta _{1} & 3 & 5\\
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2 & 4 & 6\\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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1 & 4\\
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2 & 5\\
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3 & 6\\
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\end{pmatrix}
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=
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\begin{pmatrix}
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20 & 41\\
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28 & \theta _{2}\\
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\end{pmatrix}
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\end{equation*}
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\begin{enumerate}[label=\Alph*)]
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\item $\theta _{1}=20, \theta _{2}=36$
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\item $\theta _{1}=1, \theta _{2}=60$
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\item $\theta _{1}=12, \theta _{2}=51$
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\item $\theta _{1}=1, \theta _{2}=36$
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\item $\theta _{1}=-1, \theta _{2}=64$
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\end{enumerate}
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\section*{Aufgabe 9 (5 Punkte)}
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Bestimmen Sie die Inverse $X^{-1}$ der Matrix
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\begin{equation*}
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X =
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\begin{pmatrix}
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2 & 4\\
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-5 & \theta\\
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\end{pmatrix}
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\end{equation*}
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\noindent
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in Abhängigkeit von $\theta$. Für welche Werte von $\theta$ existiert die Inverse der Matrix $X$?
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Berechnen Sie auch die Summe der beiden Hauptdiagonalelemente der inversen Matrix, d.h. die Spur der inversen Matrix $tr(X^{-1})$.
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\begin{enumerate}[label=\Alph*)]
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\item $\theta \neq 10, tr(X^{-1}) = \frac{2+\theta}{2\theta-20}$
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\item $\theta \neq -10, tr(X^{-1}) = 2 + \theta$
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\item $\theta \neq 5, tr(X^{-1}) = \frac{\theta}{\theta - 5}$
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\item $\theta \neq -10, tr(X^{-1}) = \frac{2 + \theta}{2\theta + 20}$
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\item $\theta \neq -5, tr(X^{-1}) = \frac{\theta - 3}{2\theta + 10}$
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\end{enumerate}
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\newpage
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\section*{Aufgabe 10 (6 Punkte)}
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Wie lautet die allgemeine Lösungsmenge des inhomogenen linearen Gleichungssystems?
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\begin{equation*}
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\begin{pmatrix}
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1 & 3 & 2 & 0\\
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0 & 3 & -3 & 3\\
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1 & 2 & 3 & -1\\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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x_{1}\\
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x_{2}\\
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x_{3}\\
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x_{4}\\
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\end{pmatrix}
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=
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\begin{pmatrix}
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0\\
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6\\
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-2\\
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\end{pmatrix}
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\end{equation*}
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\begin{enumerate}[label=\Alph*)]
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\item $\mathbb{L} = \theta$
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\item $\mathbb{L} =
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\left\{
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\begin{pmatrix}
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x_{1}\\
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x_{2}\\
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x_{3}\\
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x_{4}\\
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\end{pmatrix}
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\vert
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\begin{pmatrix}
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x_{1}\\
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x_{2}\\
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x_{3}\\
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x_{4}\\
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\end{pmatrix}
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=
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\begin{pmatrix}
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-6\\
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2\\
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0\\
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0\\
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\end{pmatrix}
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+r_{1}\cdot
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\begin{pmatrix}
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-5\\
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1\\
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1\\
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0\\
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\end{pmatrix}
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+r_{2}\cdot
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\begin{pmatrix}
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3\\
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-1\\
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0\\
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1\\
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\end{pmatrix}
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; \quad r_{1},2_{2} \in \mathbb{R}
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\right\}
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$
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\item $\mathbb{L} =
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\left\{
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\begin{pmatrix}
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x_{1}\\
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x_{2}\\
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x_{3}\\
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x_{4}\\
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\end{pmatrix}
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=
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\begin{pmatrix}
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-6\\
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2\\
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4\\
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-2\\
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\end{pmatrix}
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\right\}
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$
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\item $\mathbb{L} =
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\left\{
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\begin{pmatrix}
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x_{1}\\
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x_{2}\\
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x_{3}\\
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x_{4}\\
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\end{pmatrix}
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\vert
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\begin{pmatrix}
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x_{1}\\
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x_{2}\\
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x_{3}\\
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x_{4}\\
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\end{pmatrix}
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=
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\begin{pmatrix}
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6\\
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-2\\
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0\\
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0\\
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\end{pmatrix}
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+r_{1}\cdot
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\begin{pmatrix}
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5\\
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-1\\
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1\\
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0\\
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\end{pmatrix}
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+r_{2}\cdot
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\begin{pmatrix}
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-3\\
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1\\
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0\\
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1\\
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\end{pmatrix}
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; \quad r_{1},2_{2} \in \mathbb{R}
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\right\}
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$
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\item $\mathbb{L} =
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\left\{
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\begin{pmatrix}
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x_{1}\\
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x_{2}\\
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x_{3}\\
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x_{4}\\
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\end{pmatrix}
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\vert
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\begin{pmatrix}
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x_{1}\\
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x_{2}\\
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x_{3}\\
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x_{4}\\
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\end{pmatrix}
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=
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\begin{pmatrix}
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-6\\
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2\\
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-2\\
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0\\
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\end{pmatrix}
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+r \cdot
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\begin{pmatrix}
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-5\\
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3\\
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1\\
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1\\
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\end{pmatrix}
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; \quad r_{1},2_{2} \in \mathbb{R}
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\right\}
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$
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\end{enumerate}
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\newpage
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\section*{Aufgabe 11 (5 Punkte)}
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Die Matrix
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\begin{equation*}
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A =
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\begin{pmatrix}
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a_{11} & a_{12} & a_{13}\\
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a_{21} & a_{22} & a_{23}\\
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a_{31} & a_{32} & a_{33}\\
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\end{pmatrix}
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\end{equation*}
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besitzt die Determinante $det(A)=10$. Betrachten Sie die Matrix
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\begin{equation*}
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A^{*} =
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\begin{pmatrix}
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a_{21} & a_{22} & a_{23}\\
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a_{11} & a_{12} & a_{13}\\
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2a_{31} & 2a_{32} & 2a_{33}\\
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\end{pmatrix}
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,
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\end{equation*}
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die durch elementare Zeilen- und Spaltenumformungen aus $A$ und
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Transponieren entstanden ist. Wie lautet die Determinante $det(A^{*})$.
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\begin{enumerate}[label=\Alph*)]
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\item $det(A^{*})=-5$
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\item $det(A^{*})=-20$
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\item $det(A^{*})=10$
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\item $det(A^{*})=20$
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\item $det(A^{*})=\frac{1}{20}$
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\end{enumerate}
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\section*{Aufgabe 12 (5 Punkte)}
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Wie lauten die Eigenwerte der Matrix
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\begin{equation*}
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A =
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\begin{pmatrix}
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7 & 0 & 0\\
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-1 & -3 & 0\\
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2 & -4 & 5\\
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\end{pmatrix}
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?
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\end{equation*}
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\begin{enumerate}[label=\Alph*)]
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\item $\lambda_{1}=-4,\lambda_{2}=4,\lambda_{3}=6$
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\item $\lambda_{1}=3,\lambda_{2}=-5,\lambda_{3}=-7$
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\item $\lambda_{1}=1,\lambda_{2}=4,\lambda_{3}=2$
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\item $\lambda_{1}=-3,\lambda_{2}=5,\lambda_{3}=7$
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\item $\lambda_{1}=2,\lambda_{2}=3,\lambda_{3}=0$
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\end{enumerate}
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\end{document}
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% \image{1}{Capture3.PNG}{DNS-Anfrage}{DNS-Anfrage}
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