Lineare Algebra Beispiele

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 4 & 5 & 9 & 1 & 0 & 0 & 0 & 700 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 4 & 5 & 9 & 1 & 0 & 0 & 0 & 700 \\ 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Schritt 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{4}{4} & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & \frac{0}{4} & \frac{0}{4} & \frac{0}{4} & \frac{700}{4} 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{4}{4} & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & \frac{0}{4} & \frac{0}{4} & \frac{0}{4} & \frac{700}{4} \\ 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Schritt 1.2

Perform the row operation

R_{2} = R_{2} - 6 R_{1}

$R_{2} = R_{2} - 6 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Schritt 1.2.1

Perform the row operation

R_{2} = R_{2} - 6 R_{1}

$R_{2} = R_{2} - 6 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 6 - 6 \cdot 1 & 1 - 6 (\frac{5}{4}) & 8 - 6 (\frac{9}{4}) & 0 - 6 (\frac{1}{4}) & 1 - 6 \cdot 0 & 0 - 6 \cdot 0 & 0 - 6 \cdot 0 & 300 - 6 \cdot 175 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 6 - 6 \cdot 1 & 1 - 6 (\frac{5}{4}) & 8 - 6 (\frac{9}{4}) & 0 - 6 (\frac{1}{4}) & 1 - 6 \cdot 0 & 0 - 6 \cdot 0 & 0 - 6 \cdot 0 & 300 - 6 \cdot 175 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Schritt 1.3

Perform the row operation

R_{3} = R_{3} - 8 R_{1}

$R_{3} = R_{3} - 8 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Schritt 1.3.1

Perform the row operation

R_{3} = R_{3} - 8 R_{1}

$R_{3} = R_{3} - 8 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 8 - 8 \cdot 1 & 5 - 8 (\frac{5}{4}) & 4 - 8 (\frac{9}{4}) & 0 - 8 (\frac{1}{4}) & 0 - 8 \cdot 0 & 1 - 8 \cdot 0 & 0 - 8 \cdot 0 & 700 - 8 \cdot 175 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 8 - 8 \cdot 1 & 5 - 8 (\frac{5}{4}) & 4 - 8 (\frac{9}{4}) & 0 - 8 (\frac{1}{4}) & 0 - 8 \cdot 0 & 1 - 8 \cdot 0 & 0 - 8 \cdot 0 & 700 - 8 \cdot 175 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.3.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Schritt 1.4

Perform the row operation

R_{4} = R_{4} + 3 R_{1}

$R_{4} = R_{4} + 3 R_{1}$ to make the entry at

4, 1

$4, 1$ a

0

$0$ .

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Schritt 1.4.1

Perform the row operation

R_{4} = R_{4} + 3 R_{1}

$R_{4} = R_{4} + 3 R_{1}$ to make the entry at

4, 1

$4, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 - 3 + 3 \cdot 1 & - 8 + 3 (\frac{5}{4}) & - 1 + 3 (\frac{9}{4}) & 0 + 3 (\frac{1}{4}) & 0 + 3 \cdot 0 & 0 + 3 \cdot 0 & 1 + 3 \cdot 0 & 0 + 3 \cdot 175 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ - 3 + 3 \cdot 1 & - 8 + 3 (\frac{5}{4}) & - 1 + 3 (\frac{9}{4}) & 0 + 3 (\frac{1}{4}) & 0 + 3 \cdot 0 & 0 + 3 \cdot 0 & 1 + 3 \cdot 0 & 0 + 3 \cdot 175 \end{array}]$

Schritt 1.4.2

Vereinfache

R_{4}

$R_{4}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Schritt 1.5

Multiply each element of

$R_{2}$ by

$- \frac{2}{13}$ to make the entry at

$2, 2$ a

$1$ .

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Schritt 1.5.1

Multiply each element of

$R_{2}$ by

$- \frac{2}{13}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ - \frac{2}{13} \cdot 0 & - \frac{2}{13} (- \frac{13}{2}) & - \frac{2}{13} (- \frac{11}{2}) & - \frac{2}{13} (- \frac{3}{2}) & - \frac{2}{13} \cdot 1 & - \frac{2}{13} \cdot 0 & - \frac{2}{13} \cdot 0 & - \frac{2}{13} \cdot - 750 \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{array}]$

Schritt 1.5.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{array}]$

Schritt 1.6

Perform the row operation

$R_{3} = R_{3} + 5 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Schritt 1.6.1

Perform the row operation

$R_{3} = R_{3} + 5 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - 14 + 5 (\frac{11}{13}) & - 2 + 5 (\frac{3}{13}) & 0 + 5 (- \frac{2}{13}) & 1 + 5 \cdot 0 & 0 + 5 \cdot 0 & - 700 + 5 (\frac{1500}{13}) \\ 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{array}]$

Schritt 1.6.2

Vereinfache

$R_{3}$ .

Schritt 1.7

Perform the row operation

$R_{4} = R_{4} + \frac{17}{4} R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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Schritt 1.7.1

Perform the row operation

$R_{4} = R_{4} + \frac{17}{4} R_{2}$ to make the entry at

$4, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & 0 & - \frac{127}{13} & - \frac{11}{13} & - \frac{10}{13} & 1 & 0 & - \frac{1600}{13} \\ 0 + \frac{17}{4} \cdot 0 & - \frac{17}{4} + \frac{17}{4} \cdot 1 & \frac{23}{4} + \frac{17}{4} \cdot \frac{11}{13} & \frac{3}{4} + \frac{17}{4} \cdot \frac{3}{13} & 0 + \frac{17}{4} (- \frac{2}{13}) & 0 + \frac{17}{4} \cdot 0 & 1 + \frac{17}{4} \cdot 0 & 525 + \frac{17}{4} \cdot \frac{1500}{13} \end{array}]$

Schritt 1.7.2

Vereinfache

$R_{4}$ .

Schritt 1.8

Multiply each element of

$R_{3}$ by

$- \frac{13}{127}$ to make the entry at

$3, 3$ a

$1$ .

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Schritt 1.8.1

Multiply each element of

$R_{3}$ by

$- \frac{13}{127}$ to make the entry at

$3, 3$ a

$1$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ - \frac{13}{127} \cdot 0 & - \frac{13}{127} \cdot 0 & - \frac{13}{127} (- \frac{127}{13}) & - \frac{13}{127} (- \frac{11}{13}) & - \frac{13}{127} (- \frac{10}{13}) & - \frac{13}{127} \cdot 1 & - \frac{13}{127} \cdot 0 & - \frac{13}{127} (- \frac{1600}{13}) \\ 0 & 0 & \frac{243}{26} & \frac{45}{26} & - \frac{17}{26} & 0 & 1 & \frac{13200}{13} \end{array}]$

Schritt 1.8.2

Vereinfache

$R_{3}$ .

Schritt 1.9

Perform the row operation

$R_{4} = R_{4} - \frac{243}{26} R_{3}$ to make the entry at

$4, 3$ a

$0$ .

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Schritt 1.9.1

Perform the row operation

$R_{4} = R_{4} - \frac{243}{26} R_{3}$ to make the entry at

$4, 3$ a

$0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & 0 & 1 & \frac{11}{127} & \frac{10}{127} & - \frac{13}{127} & 0 & \frac{1600}{127} \\ 0 - \frac{243}{26} \cdot 0 & 0 - \frac{243}{26} \cdot 0 & \frac{243}{26} - \frac{243}{26} \cdot 1 & \frac{45}{26} - \frac{243}{26} \cdot \frac{11}{127} & - \frac{17}{26} - \frac{243}{26} \cdot \frac{10}{127} & 0 - \frac{243}{26} (- \frac{13}{127}) & 1 - \frac{243}{26} \cdot 0 & \frac{13200}{13} - \frac{243}{26} \cdot \frac{1600}{127} \end{array}]$

Schritt 1.9.2

Vereinfache

$R_{4}$ .

Schritt 1.10

Multiply each element of

$R_{4}$ by

$\frac{127}{117}$ to make the entry at

$4, 4$ a

$1$ .

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Schritt 1.10.1

Multiply each element of

$R_{4}$ by

$\frac{127}{117}$ to make the entry at

$4, 4$ a

$1$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & 0 & 1 & \frac{11}{127} & \frac{10}{127} & - \frac{13}{127} & 0 & \frac{1600}{127} \\ \frac{127}{117} \cdot 0 & \frac{127}{117} \cdot 0 & \frac{127}{117} \cdot 0 & \frac{127}{117} \cdot \frac{117}{127} & \frac{127}{117} (- \frac{353}{254}) & \frac{127}{117} \cdot \frac{243}{254} & \frac{127}{117} \cdot 1 & \frac{127}{117} \cdot \frac{114000}{127} \end{array}]$

Schritt 1.10.2

Vereinfache

$R_{4}$ .

Schritt 1.11

Perform the row operation

$R_{3} = R_{3} - \frac{11}{127} R_{4}$ to make the entry at

$3, 4$ a

$0$ .

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Schritt 1.11.1

Perform the row operation

$R_{3} = R_{3} - \frac{11}{127} R_{4}$ to make the entry at

$3, 4$ a

$0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 - \frac{11}{127} \cdot 0 & 0 - \frac{11}{127} \cdot 0 & 1 - \frac{11}{127} \cdot 0 & \frac{11}{127} - \frac{11}{127} \cdot 1 & \frac{10}{127} - \frac{11}{127} (- \frac{353}{234}) & - \frac{13}{127} - \frac{11}{127} \cdot \frac{27}{26} & 0 - \frac{11}{127} \cdot \frac{127}{117} & \frac{1600}{127} - \frac{11}{127} \cdot \frac{38000}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.11.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.12

Perform the row operation

$R_{2} = R_{2} - \frac{3}{13} R_{4}$ to make the entry at

$2, 4$ a

$0$ .

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Schritt 1.12.1

Perform the row operation

$R_{2} = R_{2} - \frac{3}{13} R_{4}$ to make the entry at

$2, 4$ a

$0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 - \frac{3}{13} \cdot 0 & 1 - \frac{3}{13} \cdot 0 & \frac{11}{13} - \frac{3}{13} \cdot 0 & \frac{3}{13} - \frac{3}{13} \cdot 1 & - \frac{2}{13} - \frac{3}{13} (- \frac{353}{234}) & 0 - \frac{3}{13} \cdot \frac{27}{26} & 0 - \frac{3}{13} \cdot \frac{127}{117} & \frac{1500}{13} - \frac{3}{13} \cdot \frac{38000}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.12.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & 0 & \frac{197}{1014} & - \frac{81}{338} & - \frac{127}{507} & - \frac{18500}{169} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.13

Perform the row operation

$R_{1} = R_{1} - \frac{1}{4} R_{4}$ to make the entry at

$1, 4$ a

$0$ .

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Schritt 1.13.1

Perform the row operation

$R_{1} = R_{1} - \frac{1}{4} R_{4}$ to make the entry at

$1, 4$ a

$0$ .

$[\begin{array}{c} 1 - \frac{1}{4} \cdot 0 & \frac{5}{4} - \frac{1}{4} \cdot 0 & \frac{9}{4} - \frac{1}{4} \cdot 0 & \frac{1}{4} - \frac{1}{4} \cdot 1 & 0 - \frac{1}{4} (- \frac{353}{234}) & 0 - \frac{1}{4} \cdot \frac{27}{26} & 0 - \frac{1}{4} \cdot \frac{127}{117} & 175 - \frac{1}{4} \cdot \frac{38000}{39} \\ 0 & 1 & \frac{11}{13} & 0 & \frac{197}{1014} & - \frac{81}{338} & - \frac{127}{507} & - \frac{18500}{169} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.13.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & 0 & \frac{353}{936} & - \frac{27}{104} & - \frac{127}{468} & - \frac{2675}{39} \\ 0 & 1 & \frac{11}{13} & 0 & \frac{197}{1014} & - \frac{81}{338} & - \frac{127}{507} & - \frac{18500}{169} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.14

Perform the row operation

$R_{2} = R_{2} - \frac{11}{13} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.14.1

Perform the row operation

$R_{2} = R_{2} - \frac{11}{13} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & 0 & \frac{353}{936} & - \frac{27}{104} & - \frac{127}{468} & - \frac{2675}{39} \\ 0 - \frac{11}{13} \cdot 0 & 1 - \frac{11}{13} \cdot 0 & \frac{11}{13} - \frac{11}{13} \cdot 1 & 0 - \frac{11}{13} \cdot 0 & \frac{197}{1014} - \frac{11}{13} \cdot \frac{49}{234} & - \frac{81}{338} - \frac{11}{13} (- \frac{5}{26}) & - \frac{127}{507} - \frac{11}{13} (- \frac{11}{117}) & - \frac{18500}{169} - \frac{11}{13} (- \frac{2800}{39}) \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.14.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & 0 & \frac{353}{936} & - \frac{27}{104} & - \frac{127}{468} & - \frac{2675}{39} \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.15

Perform the row operation

$R_{1} = R_{1} - \frac{9}{4} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.15.1

Perform the row operation

$R_{1} = R_{1} - \frac{9}{4} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

$[\begin{array}{c} 1 - \frac{9}{4} \cdot 0 & \frac{5}{4} - \frac{9}{4} \cdot 0 & \frac{9}{4} - \frac{9}{4} \cdot 1 & 0 - \frac{9}{4} \cdot 0 & \frac{353}{936} - \frac{9}{4} \cdot \frac{49}{234} & - \frac{27}{104} - \frac{9}{4} (- \frac{5}{26}) & - \frac{127}{468} - \frac{9}{4} (- \frac{11}{117}) & - \frac{2675}{39} - \frac{9}{4} (- \frac{2800}{39}) \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.15.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & 0 & 0 & - \frac{11}{117} & \frac{9}{52} & - \frac{7}{117} & \frac{3625}{39} \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.16

Perform the row operation

$R_{1} = R_{1} - \frac{5}{4} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Schritt 1.16.1

Perform the row operation

$R_{1} = R_{1} - \frac{5}{4} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - \frac{5}{4} \cdot 0 & \frac{5}{4} - \frac{5}{4} \cdot 1 & 0 - \frac{5}{4} \cdot 0 & 0 - \frac{5}{4} \cdot 0 & - \frac{11}{117} - \frac{5}{4} \cdot \frac{2}{117} & \frac{9}{52} - \frac{5}{4} (- \frac{1}{13}) & - \frac{7}{117} - \frac{5}{4} (- \frac{20}{117}) & \frac{3625}{39} - \frac{5}{4} (- \frac{1900}{39}) \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Schritt 1.16.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & - \frac{3}{26} & \frac{7}{26} & \frac{2}{13} & \frac{2000}{13} \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

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:

$a_{11}, a_{22}, a_{33},$ and

$a_{44}$

Pivot Columns:

$1, 2, 3,$ and

$4$