Lineare Algebra Beispiele

$[\begin{array}{c} 2 & 3 & 3 & - 1 & - 4 & 2 & 5 & 0 & 1 & 4 \\ 1 & 5 & 2 & 0 & 0 & 0 & 1 & - 2 & - 3 & - 4 \end{array}]$

Schritt 1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Schritt 1.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{3}{2} & \frac{3}{2} & \frac{- 1}{2} & \frac{- 4}{2} & \frac{2}{2} & \frac{5}{2} & \frac{0}{2} & \frac{1}{2} & \frac{4}{2} \\ 1 & 5 & 2 & 0 & 0 & 0 & 1 & - 2 & - 3 & - 4 \end{array}]$

Schritt 1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & - 2 & 1 & \frac{5}{2} & 0 & \frac{1}{2} & 2 \\ 1 & 5 & 2 & 0 & 0 & 0 & 1 & - 2 & - 3 & - 4 \end{array}]$

Schritt 2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & - 2 & 1 & \frac{5}{2} & 0 & \frac{1}{2} & 2 \\ 1 - 1 & 5 - \frac{3}{2} & 2 - \frac{3}{2} & 0 + \frac{1}{2} & 0 + 2 & 0 - 1 & 1 - \frac{5}{2} & - 2 - 0 & - 3 - \frac{1}{2} & - 4 - 2 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & - 2 & 1 & \frac{5}{2} & 0 & \frac{1}{2} & 2 \\ 0 & \frac{7}{2} & \frac{1}{2} & \frac{1}{2} & 2 & - 1 & - \frac{3}{2} & - 2 & - \frac{7}{2} & - 6 \end{array}]$

Schritt 3

Multiply each element of $R_{2}$ by $\frac{2}{7}$ to make the entry at $2, 2$ a $1$ .

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Schritt 3.1

Multiply each element of $R_{2}$ by $\frac{2}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & - 2 & 1 & \frac{5}{2} & 0 & \frac{1}{2} & 2 \\ \frac{2}{7} \cdot 0 & \frac{2}{7} \cdot \frac{7}{2} & \frac{2}{7} \cdot \frac{1}{2} & \frac{2}{7} \cdot \frac{1}{2} & \frac{2}{7} \cdot 2 & \frac{2}{7} \cdot - 1 & \frac{2}{7} (- \frac{3}{2}) & \frac{2}{7} \cdot - 2 & \frac{2}{7} (- \frac{7}{2}) & \frac{2}{7} \cdot - 6 \end{array}]$

Schritt 3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & - 2 & 1 & \frac{5}{2} & 0 & \frac{1}{2} & 2 \\ 0 & 1 & \frac{1}{7} & \frac{1}{7} & \frac{4}{7} & - \frac{2}{7} & - \frac{3}{7} & - \frac{4}{7} & - 1 & - \frac{12}{7} \end{array}]$

Schritt 4

Perform the row operation $R_{1} = R_{1} - \frac{3}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 4.1

Perform the row operation $R_{1} = R_{1} - \frac{3}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & \frac{3}{2} - \frac{3}{2} \cdot \frac{1}{7} & - \frac{1}{2} - \frac{3}{2} \cdot \frac{1}{7} & - 2 - \frac{3}{2} \cdot \frac{4}{7} & 1 - \frac{3}{2} (- \frac{2}{7}) & \frac{5}{2} - \frac{3}{2} (- \frac{3}{7}) & 0 - \frac{3}{2} (- \frac{4}{7}) & \frac{1}{2} - \frac{3}{2} \cdot - 1 & 2 - \frac{3}{2} (- \frac{12}{7}) \\ 0 & 1 & \frac{1}{7} & \frac{1}{7} & \frac{4}{7} & - \frac{2}{7} & - \frac{3}{7} & - \frac{4}{7} & - 1 & - \frac{12}{7} \end{array}]$

Schritt 4.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{9}{7} & - \frac{5}{7} & - \frac{20}{7} & \frac{10}{7} & \frac{22}{7} & \frac{6}{7} & 2 & \frac{32}{7} \\ 0 & 1 & \frac{1}{7} & \frac{1}{7} & \frac{4}{7} & - \frac{2}{7} & - \frac{3}{7} & - \frac{4}{7} & - 1 & - \frac{12}{7} \end{array}]$