Lineare Algebra Beispiele

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 1 & 0.2 & 0.8 & 428 \\ 3.67 & 3.86 & 0.8 & 1610.9 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 1 - 1 & 0.2 - 1.61 & 0.8 - 0.35 & 428 - 564.2 \\ 3.67 & 3.86 & 0.8 & 1610.9 \end{array}]$

Schritt 1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & - 1.41 & 0.45 & - 136.2 \\ 3.67 & 3.86 & 0.8 & 1610.9 \end{array}]$

Schritt 2

Perform the row operation $R_{3} = R_{3} - 3.67 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 2.1

Perform the row operation $R_{3} = R_{3} - 3.67 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & - 1.41 & 0.45 & - 136.2 \\ 3.67 - 3.67 \cdot 1 & 3.86 - 3.67 \cdot 1.61 & 0.8 - 3.67 \cdot 0.35 & 1610.9 - 3.67 \cdot 564.2 \end{array}]$

Schritt 2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & - 1.41 & 0.45 & - 136.2 \\ 0 & - 2.0487 & - 0.4845 & - 459.714 \end{array}]$

Schritt 3

Multiply each element of $R_{2}$ by $\frac{1}{- 1.41}$ to make the entry at $2, 2$ a $1$ .

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Schritt 3.1

Multiply each element of $R_{2}$ by $\frac{1}{- 1.41}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ \frac{0}{- 1.41} & \frac{- 1.41}{- 1.41} & \frac{0.45}{- 1.41} & \frac{- 136.2}{- 1.41} \\ 0 & - 2.0487 & - 0.4845 & - 459.714 \end{array}]$

Schritt 3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & 1 & - 0.31914893 & 96.59574468 \\ 0 & - 2.0487 & - 0.4845 & - 459.714 \end{array}]$

Schritt 4

Perform the row operation $R_{3} = R_{3} + 2.0487 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 4.1

Perform the row operation $R_{3} = R_{3} + 2.0487 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & 1 & - 0.31914893 & 96.59574468 \\ 0 + 2.0487 \cdot 0 & - 2.0487 + 2.0487 \cdot 1 & - 0.4845 + 2.0487 \cdot - 0.31914893 & - 459.714 + 2.0487 \cdot 96.59574468 \end{array}]$

Schritt 4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & 1 & - 0.31914893 & 96.59574468 \\ 0 & 0 & - 1.13834042 & - 261.81829787 \end{array}]$

Schritt 5

Multiply each element of $R_{3}$ by $\frac{1}{- 1.13834042}$ to make the entry at $3, 3$ a $1$ .

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Schritt 5.1

Multiply each element of $R_{3}$ by $\frac{1}{- 1.13834042}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & 1 & - 0.31914893 & 96.59574468 \\ \frac{0}{- 1.13834042} & \frac{0}{- 1.13834042} & \frac{- 1.13834042}{- 1.13834042} & \frac{- 261.81829787}{- 1.13834042} \end{array}]$

Schritt 5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & 1 & - 0.31914893 & 96.59574468 \\ 0 & 0 & 1 & 230 \end{array}]$

Schritt 6

Perform the row operation $R_{2} = R_{2} + 0.31914893 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 6.1

Perform the row operation $R_{2} = R_{2} + 0.31914893 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 + 0.31914893 \cdot 0 & 1 + 0.31914893 \cdot 0 & - 0.31914893 + 0.31914893 \cdot 1 & 96.59574468 + 0.31914893 \cdot 230 \\ 0 & 0 & 1 & 230 \end{array}]$

Schritt 6.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1.61 & 0.35 & 564.2 \\ 0 & 1 & 0 & 170 \\ 0 & 0 & 1 & 230 \end{array}]$

Schritt 7

Perform the row operation $R_{1} = R_{1} - 0.35 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 7.1

Perform the row operation $R_{1} = R_{1} - 0.35 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0.35 \cdot 0 & 1.61 - 0.35 \cdot 0 & 0.35 - 0.35 \cdot 1 & 564.2 - 0.35 \cdot 230 \\ 0 & 1 & 0 & 170 \\ 0 & 0 & 1 & 230 \end{array}]$

Schritt 7.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1.61 & 0 & 483.7 \\ 0 & 1 & 0 & 170 \\ 0 & 0 & 1 & 230 \end{array}]$

Schritt 8

Perform the row operation $R_{1} = R_{1} - 1.61 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 8.1

Perform the row operation $R_{1} = R_{1} - 1.61 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 1.61 \cdot 0 & 1.61 - 1.61 \cdot 1 & 0 - 1.61 \cdot 0 & 483.7 - 1.61 \cdot 170 \\ 0 & 1 & 0 & 170 \\ 0 & 0 & 1 & 230 \end{array}]$

Schritt 8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 210 \\ 0 & 1 & 0 & 170 \\ 0 & 0 & 1 & 230 \end{array}]$