Finite Mathematik Beispiele

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0.9 & 1 & 1.1 & 1000 \\ 0 & 1 & - 1 & 2260 \end{array}]$

Schritt 1

Perform the row operation $R_{2} = R_{2} - 0.9 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 1.1

Perform the row operation $R_{2} = R_{2} - 0.9 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0.9 - 0.9 \cdot 1 & 1 - 0.9 \cdot 1 & 1.1 - 0.9 \cdot 1 & 1000 - 0.9 \cdot 23000 \\ 0 & 1 & - 1 & 2260 \end{array}]$

Schritt 1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0 & 0.1 & 0.2 & - 19700 \\ 0 & 1 & - 1 & 2260 \end{array}]$

Schritt 2

Multiply each element of $R_{2}$ by $\frac{1}{0.1}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.1

Multiply each element of $R_{2}$ by $\frac{1}{0.1}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ \frac{0}{0.1} & \frac{0.1}{0.1} & \frac{0.2}{0.1} & \frac{- 19700}{0.1} \\ 0 & 1 & - 1 & 2260 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0 & 1 & 2 & - 197000 \\ 0 & 1 & - 1 & 2260 \end{array}]$

Schritt 3

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

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Schritt 3.1

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

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0 & 1 & 2 & - 197000 \\ 0 - 0 & 1 - 1 & - 1 - 2 & 2260 + 197000 \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0 & 1 & 2 & - 197000 \\ 0 & 0 & - 3 & 199260 \end{array}]$

Schritt 4

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

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Schritt 4.1

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

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0 & 1 & 2 & - 197000 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 199260 \end{array}]$

Schritt 4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0 & 1 & 2 & - 197000 \\ 0 & 0 & 1 & - 66420 \end{array}]$

Schritt 5

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

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Schritt 5.1

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

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 197000 - 2 \cdot - 66420 \\ 0 & 0 & 1 & - 66420 \end{array}]$

Schritt 5.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 23000 \\ 0 & 1 & 0 & - 64160 \\ 0 & 0 & 1 & - 66420 \end{array}]$

Schritt 6

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

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Schritt 6.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 23000 + 66420 \\ 0 & 1 & 0 & - 64160 \\ 0 & 0 & 1 & - 66420 \end{array}]$

Schritt 6.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 89420 \\ 0 & 1 & 0 & - 64160 \\ 0 & 0 & 1 & - 66420 \end{array}]$

Schritt 7

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

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Schritt 7.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 89420 + 64160 \\ 0 & 1 & 0 & - 64160 \\ 0 & 0 & 1 & - 66420 \end{array}]$

Schritt 7.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 153580 \\ 0 & 1 & 0 & - 64160 \\ 0 & 0 & 1 & - 66420 \end{array}]$