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86 lines
1.7 KiB
TeX
86 lines
1.7 KiB
TeX
\section*{Aufgabe 1}
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\textbf{Gegeben:}
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\[A =
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\begin{pmatrix}
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6 & -6 & 0 \\
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-3 & 7 & 2 \\
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2 & 4 & 1
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\end{pmatrix}, \;\;\;
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b =\begin{pmatrix}
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0 \\
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8 \\
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8
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\end{pmatrix}\]
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\textbf{Aufgabe:} $Ax = b$ mit Gaußelimination und Spaltenpivotwahl lösen
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\textbf{Lösung:}
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\begin{align}
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\begin{gmatrix}[p]
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6 & -6 & 0 & 0\\
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-3 & 7 & 2 & 8\\
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2 & 4 & 1 & 8
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\rowops
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\mult{0}{\cdot \frac{1}{6}}
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\end{gmatrix}
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&\leadsto
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\begin{gmatrix}[p]
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1 & -1 & 0 & 0\\
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-3 & 7 & 2 & 8\\
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2 & 4 & 1 & 8
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\rowops
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\add[3]{0}{1}
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\add[-2]{0}{2}
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\end{gmatrix}\\
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&\leadsto
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\begin{gmatrix}[p]
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1 & -1 & 0 & 0\\
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0 & 4 & 2 & 8\\
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0 & 6 & 1 & 8
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\rowops
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\swap{1}{2}
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\end{gmatrix}\\
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&\leadsto
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\begin{gmatrix}[p]
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1 & -1 & 0 & 0\\
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0 & 6 & 1 & 8\\
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0 & 4 & 2 & 8
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\rowops
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\mult{1}{\cdot \frac{1}{6}}
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\add[-4]{1}{2}
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\end{gmatrix}\\
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&\leadsto
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\begin{gmatrix}[p]
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1 & -1 & 0 & 0\\
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0 & 1 & \frac{1}{6} & \frac{4}{3}\\
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0 & 0 & \frac{4}{3} & \frac{8}{3}
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\rowops
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\mult{2}{\cdot \frac{3}{4}}
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\end{gmatrix}\\
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&\leadsto
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\begin{gmatrix}[p]
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1 & -1 & 0 & 0\\
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0 & 1 & \frac{1}{6} & \frac{4}{3}\\
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0 & 0 & 1 & 2
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\rowops
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\add[\cdot \frac{-1}{6}]{2}{1}
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\end{gmatrix}\\
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&\leadsto
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\begin{gmatrix}[p]
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1 & -1 & 0 & 0\\
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0 & 1 & 0 & 1\\
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0 & 0 & 1 & 2
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\rowops
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\add[]{1}{0}
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\end{gmatrix}\\
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&\leadsto
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\begin{gmatrix}[p]
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1 & 0 & 0 & 1\\
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0 & 1 & 0 & 1\\
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0 & 0 & 1 & 2
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\end{gmatrix}
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\end{align}
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$\Rightarrow x = \begin{pmatrix}1 & 1 & 2\end{pmatrix}^T$ löst das LGS.
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