Lineare Algebra Beispiele

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 1 & 2 & - 3 & 0 & - 5 & 0 \\ 1 & 3 & - 4 & 0 & - 13 & - 11 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Schritt 1

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

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Schritt 1.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 1 - 1 & 2 - 1 & - 3 + 2 & 0 - 0 & - 5 + 3 & 0 + 1 \\ 1 & 3 & - 4 & 0 & - 13 & - 11 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Schritt 1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 1 & 3 & - 4 & 0 & - 13 & - 11 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Schritt 2

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

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Schritt 2.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 1 - 1 & 3 - 1 & - 4 + 2 & 0 - 0 & - 13 + 3 & - 11 + 1 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Schritt 2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Schritt 3

Perform the row operation $R_{4} = R_{4} - R_{1}$ to make the entry at $4, 1$ a $0$ .

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Schritt 3.1

Perform the row operation $R_{4} = R_{4} - R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 1 - 1 & - 1 - 1 & 1 + 2 & 1 - 0 & - 10 + 3 & 0 + 1 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Schritt 3.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Schritt 4

Perform the row operation $R_{5} = R_{5} - R_{1}$ to make the entry at $5, 1$ a $0$ .

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Schritt 4.1

Perform the row operation $R_{5} = R_{5} - R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 1 - 1 & 4 - 1 & - 5 + 2 & 0 - 0 & - 1 + 3 & 23 + 1 \end{array}]$

Schritt 4.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Schritt 5

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

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Schritt 5.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 2 - 2 \cdot - 1 & 0 - 2 \cdot 0 & - 10 - 2 \cdot - 2 & - 10 - 2 \cdot 1 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Schritt 5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Schritt 6

Perform the row operation $R_{4} = R_{4} + 2 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Schritt 6.1

Perform the row operation $R_{4} = R_{4} + 2 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 3 + 2 \cdot - 1 & 1 + 2 \cdot 0 & - 7 + 2 \cdot - 2 & 1 + 2 \cdot 1 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Schritt 6.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Schritt 7

Perform the row operation $R_{5} = R_{5} - 3 R_{2}$ to make the entry at $5, 2$ a $0$ .

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Schritt 7.1

Perform the row operation $R_{5} = R_{5} - 3 R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & - 3 - 3 \cdot - 1 & 0 - 3 \cdot 0 & 2 - 3 \cdot - 2 & 24 - 3 \cdot 1 \end{array}]$

Schritt 7.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 8 & 21 \end{array}]$

Schritt 8

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & 0 & 0 & 0 & 8 & 21 \end{array}]$

Schritt 9

Multiply each element of $R_{4}$ by $- \frac{1}{6}$ to make the entry at $4, 5$ a $1$ .

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Schritt 9.1

Multiply each element of $R_{4}$ by $- \frac{1}{6}$ to make the entry at $4, 5$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 12 \\ 0 & 0 & 0 & 0 & 8 & 21 \end{array}]$

Schritt 9.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 8 & 21 \end{array}]$

Schritt 10

Perform the row operation $R_{5} = R_{5} - 8 R_{4}$ to make the entry at $5, 5$ a $0$ .

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Schritt 10.1

Perform the row operation $R_{5} = R_{5} - 8 R_{4}$ to make the entry at $5, 5$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 - 8 \cdot 0 & 0 - 8 \cdot 0 & 0 - 8 \cdot 0 & 0 - 8 \cdot 0 & 8 - 8 \cdot 1 & 21 - 8 \cdot 2 \end{array}]$

Schritt 10.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 5 \end{array}]$

Schritt 11

Multiply each element of $R_{5}$ by $\frac{1}{5}$ to make the entry at $5, 6$ a $1$ .

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Schritt 11.1

Multiply each element of $R_{5}$ by $\frac{1}{5}$ to make the entry at $5, 6$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ \frac{0}{5} & \frac{0}{5} & \frac{0}{5} & \frac{0}{5} & \frac{0}{5} & \frac{5}{5} \end{array}]$

Schritt 11.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 12

Perform the row operation $R_{4} = R_{4} - 2 R_{5}$ to make the entry at $4, 6$ a $0$ .

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Schritt 12.1

Perform the row operation $R_{4} = R_{4} - 2 R_{5}$ to make the entry at $4, 6$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 12.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 13

Perform the row operation $R_{3} = R_{3} - 3 R_{5}$ to make the entry at $3, 6$ a $0$ .

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Schritt 13.1

Perform the row operation $R_{3} = R_{3} - 3 R_{5}$ to make the entry at $3, 6$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 - 3 \cdot 0 & 0 - 3 \cdot 0 & 1 - 3 \cdot 0 & 1 - 3 \cdot 0 & - 11 - 3 \cdot 0 & 3 - 3 \cdot 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 13.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 14

Perform the row operation $R_{2} = R_{2} - R_{5}$ to make the entry at $2, 6$ a $0$ .

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Schritt 14.1

Perform the row operation $R_{2} = R_{2} - R_{5}$ to make the entry at $2, 6$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 - 0 & 1 - 0 & - 1 - 0 & 0 - 0 & - 2 - 0 & 1 - 1 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 14.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 0 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 15

Perform the row operation $R_{1} = R_{1} + R_{5}$ to make the entry at $1, 6$ a $0$ .

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Schritt 15.1

Perform the row operation $R_{1} = R_{1} + R_{5}$ to make the entry at $1, 6$ a $0$ .

$[\begin{array}{c} 1 + 0 & 1 + 0 & - 2 + 0 & 0 + 0 & - 3 + 0 & - 1 + 1 \cdot 1 \\ 0 & 1 & - 1 & 0 & - 2 & 0 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 15.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & 0 \\ 0 & 1 & - 1 & 0 & - 2 & 0 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 16

Perform the row operation $R_{3} = R_{3} + 11 R_{4}$ to make the entry at $3, 5$ a $0$ .

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Schritt 16.1

Perform the row operation $R_{3} = R_{3} + 11 R_{4}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & 0 \\ 0 & 1 & - 1 & 0 & - 2 & 0 \\ 0 + 11 \cdot 0 & 0 + 11 \cdot 0 & 1 + 11 \cdot 0 & 1 + 11 \cdot 0 & - 11 + 11 \cdot 1 & 0 + 11 \cdot 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 16.2

Vereinfache $R_{3}$ .

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

Schritt 17

Perform the row operation $R_{2} = R_{2} + 2 R_{4}$ to make the entry at $2, 5$ a $0$ .

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Schritt 17.1

Perform the row operation $R_{2} = R_{2} + 2 R_{4}$ to make the entry at $2, 5$ a $0$ .

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

Schritt 17.2

Vereinfache $R_{2}$ .

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

Schritt 18

Perform the row operation $R_{1} = R_{1} + 3 R_{4}$ to make the entry at $1, 5$ a $0$ .

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Schritt 18.1

Perform the row operation $R_{1} = R_{1} + 3 R_{4}$ to make the entry at $1, 5$ a $0$ .

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

Schritt 18.2

Vereinfache $R_{1}$ .

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

Schritt 19

Perform the row operation $R_{2} = R_{2} + R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 19.1

Perform the row operation $R_{2} = R_{2} + R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & 0 & 0 \\ 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 0 + 1 \cdot 1 & 0 + 0 & 0 + 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 19.2

Vereinfache $R_{2}$ .

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

Schritt 20

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

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Schritt 20.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 1 & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 20.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 2 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Schritt 21

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

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Schritt 21.1

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

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

Schritt 21.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$