Lineare Algebra Beispiele

$[\begin{array}{c} 3 & 7 & 8 & 90 \\ 2 & - 2 & 2 & 10 \\ 4 & 7 & 8 & 92 \\ 3 & 4 & 6 & 64 \end{array}]$

Schritt 1

Multiply each element of $R_{1}$ by $\frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

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Schritt 1.1

Multiply each element of $R_{1}$ by $\frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{7}{3} & \frac{8}{3} & \frac{90}{3} \\ 2 & - 2 & 2 & 10 \\ 4 & 7 & 8 & 92 \\ 3 & 4 & 6 & 64 \end{array}]$

Schritt 1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 2 & - 2 & 2 & 10 \\ 4 & 7 & 8 & 92 \\ 3 & 4 & 6 & 64 \end{array}]$

Schritt 2

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

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Schritt 2.1

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

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 2 - 2 \cdot 1 & - 2 - 2 (\frac{7}{3}) & 2 - 2 (\frac{8}{3}) & 10 - 2 \cdot 30 \\ 4 & 7 & 8 & 92 \\ 3 & 4 & 6 & 64 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & - \frac{20}{3} & - \frac{10}{3} & - 50 \\ 4 & 7 & 8 & 92 \\ 3 & 4 & 6 & 64 \end{array}]$

Schritt 3

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

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Schritt 3.1

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

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & - \frac{20}{3} & - \frac{10}{3} & - 50 \\ 4 - 4 \cdot 1 & 7 - 4 (\frac{7}{3}) & 8 - 4 (\frac{8}{3}) & 92 - 4 \cdot 30 \\ 3 & 4 & 6 & 64 \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & - \frac{20}{3} & - \frac{10}{3} & - 50 \\ 0 & - \frac{7}{3} & - \frac{8}{3} & - 28 \\ 3 & 4 & 6 & 64 \end{array}]$

Schritt 4

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

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Schritt 4.1

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

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & - \frac{20}{3} & - \frac{10}{3} & - 50 \\ 0 & - \frac{7}{3} & - \frac{8}{3} & - 28 \\ 3 - 3 \cdot 1 & 4 - 3 (\frac{7}{3}) & 6 - 3 (\frac{8}{3}) & 64 - 3 \cdot 30 \end{array}]$

Schritt 4.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & - \frac{20}{3} & - \frac{10}{3} & - 50 \\ 0 & - \frac{7}{3} & - \frac{8}{3} & - 28 \\ 0 & - 3 & - 2 & - 26 \end{array}]$

Schritt 5

Multiply each element of $R_{2}$ by $- \frac{3}{20}$ to make the entry at $2, 2$ a $1$ .

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Schritt 5.1

Multiply each element of $R_{2}$ by $- \frac{3}{20}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ - \frac{3}{20} \cdot 0 & - \frac{3}{20} (- \frac{20}{3}) & - \frac{3}{20} (- \frac{10}{3}) & - \frac{3}{20} \cdot - 50 \\ 0 & - \frac{7}{3} & - \frac{8}{3} & - 28 \\ 0 & - 3 & - 2 & - 26 \end{array}]$

Schritt 5.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ 0 & - \frac{7}{3} & - \frac{8}{3} & - 28 \\ 0 & - 3 & - 2 & - 26 \end{array}]$

Schritt 6

Perform the row operation $R_{3} = R_{3} + \frac{7}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 6.1

Perform the row operation $R_{3} = R_{3} + \frac{7}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ 0 + \frac{7}{3} \cdot 0 & - \frac{7}{3} + \frac{7}{3} \cdot 1 & - \frac{8}{3} + \frac{7}{3} \cdot \frac{1}{2} & - 28 + \frac{7}{3} \cdot \frac{15}{2} \\ 0 & - 3 & - 2 & - 26 \end{array}]$

Schritt 6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ 0 & 0 & - \frac{3}{2} & - \frac{21}{2} \\ 0 & - 3 & - 2 & - 26 \end{array}]$

Schritt 7

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

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Schritt 7.1

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

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ 0 & 0 & - \frac{3}{2} & - \frac{21}{2} \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 2 + 3 (\frac{1}{2}) & - 26 + 3 (\frac{15}{2}) \end{array}]$

Schritt 7.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ 0 & 0 & - \frac{3}{2} & - \frac{21}{2} \\ 0 & 0 & - \frac{1}{2} & - \frac{7}{2} \end{array}]$

Schritt 8

Multiply each element of $R_{3}$ by $- \frac{2}{3}$ to make the entry at $3, 3$ a $1$ .

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Schritt 8.1

Multiply each element of $R_{3}$ by $- \frac{2}{3}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ - \frac{2}{3} \cdot 0 & - \frac{2}{3} \cdot 0 & - \frac{2}{3} (- \frac{3}{2}) & - \frac{2}{3} (- \frac{21}{2}) \\ 0 & 0 & - \frac{1}{2} & - \frac{7}{2} \end{array}]$

Schritt 8.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ 0 & 0 & 1 & 7 \\ 0 & 0 & - \frac{1}{2} & - \frac{7}{2} \end{array}]$

Schritt 9

Perform the row operation $R_{4} = R_{4} + \frac{1}{2} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Schritt 9.1

Perform the row operation $R_{4} = R_{4} + \frac{1}{2} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ 0 & 0 & 1 & 7 \\ 0 + \frac{1}{2} \cdot 0 & 0 + \frac{1}{2} \cdot 0 & - \frac{1}{2} + \frac{1}{2} \cdot 1 & - \frac{7}{2} + \frac{1}{2} \cdot 7 \end{array}]$

Schritt 9.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & \frac{1}{2} & \frac{15}{2} \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 10

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

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Schritt 10.1

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

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 - \frac{1}{2} \cdot 0 & 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & \frac{15}{2} - \frac{1}{2} \cdot 7 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 10.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & \frac{8}{3} & 30 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 11

Perform the row operation $R_{1} = R_{1} - \frac{8}{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 11.1

Perform the row operation $R_{1} = R_{1} - \frac{8}{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{8}{3} \cdot 0 & \frac{7}{3} - \frac{8}{3} \cdot 0 & \frac{8}{3} - \frac{8}{3} \cdot 1 & 30 - \frac{8}{3} \cdot 7 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 11.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{7}{3} & 0 & \frac{34}{3} \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 12

Perform the row operation $R_{1} = R_{1} - \frac{7}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 12.1

Perform the row operation $R_{1} = R_{1} - \frac{7}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{7}{3} \cdot 0 & \frac{7}{3} - \frac{7}{3} \cdot 1 & 0 - \frac{7}{3} \cdot 0 & \frac{34}{3} - \frac{7}{3} \cdot 4 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 12.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \end{array}]$