Finite Mathematik Beispiele

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 1 & 3 & 0 & 1 & 0 & 0 & 0 & 102 \\ 1 & 4 & 0 & 0 & 1 & 0 & 0 & 96 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ - 2 & - 5 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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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 $0$ .

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Schritt 1.1.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 1 - 1 & 3 - 2 & 0 - 1 & 1 - 0 & 0 - 0 & 0 - 0 & 0 - 0 & 102 - 120 \\ 1 & 4 & 0 & 0 & 1 & 0 & 0 & 96 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ - 2 & - 5 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 1 & 4 & 0 & 0 & 1 & 0 & 0 & 96 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ - 2 & - 5 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.2

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

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Schritt 1.2.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 1 - 1 & 4 - 2 & 0 - 1 & 0 - 0 & 1 - 0 & 0 - 0 & 0 - 0 & 96 - 120 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ - 2 & - 5 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 2 & - 1 & 0 & 1 & 0 & 0 & - 24 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ - 2 & - 5 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.3

Perform the row operation $R_{5} = R_{5} + 2 R_{1}$ to make the entry at $5, 1$ a $0$ .

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Schritt 1.3.1

Perform the row operation $R_{5} = R_{5} + 2 R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 2 & - 1 & 0 & 1 & 0 & 0 & - 24 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ - 2 + 2 \cdot 1 & - 5 + 2 \cdot 2 & 0 + 2 \cdot 1 & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & 0 + 2 \cdot 120 \end{array}]$

Schritt 1.3.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 2 & - 1 & 0 & 1 & 0 & 0 & - 24 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ 0 & - 1 & 2 & 0 & 0 & 0 & 1 & 240 \end{array}]$

Schritt 1.4

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 1.4.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 1 - 2 \cdot - 1 & 0 - 2 \cdot 1 & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & - 24 - 2 \cdot - 18 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ 0 & - 1 & 2 & 0 & 0 & 0 & 1 & 240 \end{array}]$

Schritt 1.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ 0 & - 1 & 2 & 0 & 0 & 0 & 1 & 240 \end{array}]$

Schritt 1.5

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

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Schritt 1.5.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 - 0 & 1 - 1 & 0 + 1 & 0 - 1 & 0 - 0 & 1 - 0 & 0 - 0 & 84 + 18 \\ 0 & - 1 & 2 & 0 & 0 & 0 & 1 & 240 \end{array}]$

Schritt 1.5.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 1 & - 1 & 0 & 1 & 0 & 102 \\ 0 & - 1 & 2 & 0 & 0 & 0 & 1 & 240 \end{array}]$

Schritt 1.6

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

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Schritt 1.6.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 1 & - 1 & 0 & 1 & 0 & 102 \\ 0 + 0 & - 1 + 1 \cdot 1 & 2 - 1 & 0 + 1 \cdot 1 & 0 + 0 & 0 + 0 & 1 + 0 & 240 - 18 \end{array}]$

Schritt 1.6.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 1 & - 1 & 0 & 1 & 0 & 102 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 222 \end{array}]$

Schritt 1.7

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

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Schritt 1.7.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 - 0 & 0 - 0 & 1 - 1 & - 1 + 2 & 0 - 1 & 1 - 0 & 0 - 0 & 102 - 12 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 222 \end{array}]$

Schritt 1.7.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 & 90 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 222 \end{array}]$

Schritt 1.8

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

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Schritt 1.8.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 & 90 \\ 0 - 0 & 0 - 0 & 1 - 1 & 1 + 2 & 0 - 1 & 0 - 0 & 1 - 0 & 222 - 12 \end{array}]$

Schritt 1.8.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 & 90 \\ 0 & 0 & 0 & 3 & - 1 & 0 & 1 & 210 \end{array}]$

Schritt 1.9

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

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Schritt 1.9.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 & 90 \\ 0 - 3 \cdot 0 & 0 - 3 \cdot 0 & 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & - 1 - 3 \cdot - 1 & 0 - 3 \cdot 1 & 1 - 3 \cdot 0 & 210 - 3 \cdot 90 \end{array}]$

Schritt 1.9.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 & 90 \\ 0 & 0 & 0 & 0 & 2 & - 3 & 1 & - 60 \end{array}]$

Schritt 1.10

Multiply each element of $R_{5}$ by $\frac{1}{2}$ to make the entry at $5, 5$ a $1$ .

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Schritt 1.10.1

Multiply each element of $R_{5}$ by $\frac{1}{2}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 & 90 \\ \frac{0}{2} & \frac{0}{2} & \frac{0}{2} & \frac{0}{2} & \frac{2}{2} & \frac{- 3}{2} & \frac{1}{2} & \frac{- 60}{2} \end{array}]$

Schritt 1.10.2

Vereinfache $R_{5}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 & 90 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.11

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

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Schritt 1.11.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 + 0 & 0 + 0 & 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 1 - \frac{3}{2} & 0 + \frac{1}{2} & 90 - 30 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.11.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.12

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

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Schritt 1.12.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 - 0 & 0 - 0 & 1 - 0 & - 2 - 0 & 1 - 1 & 0 + \frac{3}{2} & 0 - \frac{1}{2} & 12 + 30 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.12.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & - 2 & 0 & \frac{3}{2} & - \frac{1}{2} & 42 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.13

Perform the row operation $R_{3} = R_{3} + 2 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Schritt 1.13.1

Perform the row operation $R_{3} = R_{3} + 2 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & \frac{3}{2} + 2 (- \frac{1}{2}) & - \frac{1}{2} + 2 (\frac{1}{2}) & 42 + 2 \cdot 60 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.13.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & - 18 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.14

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

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Schritt 1.14.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 - 0 & 1 - 0 & - 1 - 0 & 1 - 1 & 0 - 0 & 0 + \frac{1}{2} & 0 - \frac{1}{2} & - 18 - 60 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.14.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & - 1 & 0 & 0 & \frac{1}{2} & - \frac{1}{2} & - 78 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.15

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

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Schritt 1.15.1

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

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 + 0 & \frac{1}{2} + \frac{1}{2} & - \frac{1}{2} + \frac{1}{2} & - 78 + 1 \cdot 162 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.15.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 0 & 0 & 0 & 0 & 120 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.16

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

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Schritt 1.16.1

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

$[\begin{array}{c} 1 - 0 & 2 - 0 & 1 - 1 & 0 - 0 & 0 - 0 & 0 - \frac{1}{2} & 0 - \frac{1}{2} & 120 - 162 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.16.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & 0 & 0 & - \frac{1}{2} & - \frac{1}{2} & - 42 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.17

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

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Schritt 1.17.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & - \frac{1}{2} - 2 \cdot 1 & - \frac{1}{2} - 2 \cdot 0 & - 42 - 2 \cdot 84 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 1.17.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & - \frac{5}{2} & - \frac{1}{2} & - 210 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 84 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 162 \\ 0 & 0 & 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{2} & 60 \\ 0 & 0 & 0 & 0 & 1 & - \frac{3}{2} & \frac{1}{2} & - 30 \end{array}]$

Schritt 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22}, a_{33}, a_{44},$ and $a_{55}$

Pivot Columns: $1, 2, 3, 4,$ and $5$