Finite Mathematik Beispiele

$[\begin{array}{c} 8 & 0 & 5 & 1 & 1 & 0 & 0 & 29 \\ 7 & 1 & 5 & 0 & 1 & 0 & 0 & 18 \\ 2 & 0 & 8 & 0 & 2 & 0 & 1 & 60 \end{array}]$

Schritt 1

Multiply each element of $R_{1}$ by $\frac{1}{8}$ to make the entry at $1, 1$ a $1$ .

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Schritt 1.1

Multiply each element of $R_{1}$ by $\frac{1}{8}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{8}{8} & \frac{0}{8} & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & \frac{0}{8} & \frac{0}{8} & \frac{29}{8} \\ 7 & 1 & 5 & 0 & 1 & 0 & 0 & 18 \\ 2 & 0 & 8 & 0 & 2 & 0 & 1 & 60 \end{array}]$

Schritt 1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 7 & 1 & 5 & 0 & 1 & 0 & 0 & 18 \\ 2 & 0 & 8 & 0 & 2 & 0 & 1 & 60 \end{array}]$

Schritt 2

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

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Schritt 2.1

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

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 7 - 7 \cdot 1 & 1 - 7 \cdot 0 & 5 - 7 (\frac{5}{8}) & 0 - 7 (\frac{1}{8}) & 1 - 7 (\frac{1}{8}) & 0 - 7 \cdot 0 & 0 - 7 \cdot 0 & 18 - 7 (\frac{29}{8}) \\ 2 & 0 & 8 & 0 & 2 & 0 & 1 & 60 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 0 & 1 & \frac{5}{8} & - \frac{7}{8} & \frac{1}{8} & 0 & 0 & - \frac{59}{8} \\ 2 & 0 & 8 & 0 & 2 & 0 & 1 & 60 \end{array}]$

Schritt 3

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

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Schritt 3.1

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

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 0 & 1 & \frac{5}{8} & - \frac{7}{8} & \frac{1}{8} & 0 & 0 & - \frac{59}{8} \\ 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 8 - 2 (\frac{5}{8}) & 0 - 2 (\frac{1}{8}) & 2 - 2 (\frac{1}{8}) & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 60 - 2 (\frac{29}{8}) \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 0 & 1 & \frac{5}{8} & - \frac{7}{8} & \frac{1}{8} & 0 & 0 & - \frac{59}{8} \\ 0 & 0 & \frac{27}{4} & - \frac{1}{4} & \frac{7}{4} & 0 & 1 & \frac{211}{4} \end{array}]$

Schritt 4

Multiply each element of $R_{3}$ by $\frac{4}{27}$ to make the entry at $3, 3$ a $1$ .

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Schritt 4.1

Multiply each element of $R_{3}$ by $\frac{4}{27}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 0 & 1 & \frac{5}{8} & - \frac{7}{8} & \frac{1}{8} & 0 & 0 & - \frac{59}{8} \\ \frac{4}{27} \cdot 0 & \frac{4}{27} \cdot 0 & \frac{4}{27} \cdot \frac{27}{4} & \frac{4}{27} (- \frac{1}{4}) & \frac{4}{27} \cdot \frac{7}{4} & \frac{4}{27} \cdot 0 & \frac{4}{27} \cdot 1 & \frac{4}{27} \cdot \frac{211}{4} \end{array}]$

Schritt 4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 0 & 1 & \frac{5}{8} & - \frac{7}{8} & \frac{1}{8} & 0 & 0 & - \frac{59}{8} \\ 0 & 0 & 1 & - \frac{1}{27} & \frac{7}{27} & 0 & \frac{4}{27} & \frac{211}{27} \end{array}]$

Schritt 5

Perform the row operation $R_{2} = R_{2} - \frac{5}{8} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 5.1

Perform the row operation $R_{2} = R_{2} - \frac{5}{8} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 0 - \frac{5}{8} \cdot 0 & 1 - \frac{5}{8} \cdot 0 & \frac{5}{8} - \frac{5}{8} \cdot 1 & - \frac{7}{8} - \frac{5}{8} (- \frac{1}{27}) & \frac{1}{8} - \frac{5}{8} \cdot \frac{7}{27} & 0 - \frac{5}{8} \cdot 0 & 0 - \frac{5}{8} \cdot \frac{4}{27} & - \frac{59}{8} - \frac{5}{8} \cdot \frac{211}{27} \\ 0 & 0 & 1 & - \frac{1}{27} & \frac{7}{27} & 0 & \frac{4}{27} & \frac{211}{27} \end{array}]$

Schritt 5.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{8} & \frac{1}{8} & \frac{1}{8} & 0 & 0 & \frac{29}{8} \\ 0 & 1 & 0 & - \frac{23}{27} & - \frac{1}{27} & 0 & - \frac{5}{54} & - \frac{331}{27} \\ 0 & 0 & 1 & - \frac{1}{27} & \frac{7}{27} & 0 & \frac{4}{27} & \frac{211}{27} \end{array}]$

Schritt 6

Perform the row operation $R_{1} = R_{1} - \frac{5}{8} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 6.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{8} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{8} \cdot 0 & 0 - \frac{5}{8} \cdot 0 & \frac{5}{8} - \frac{5}{8} \cdot 1 & \frac{1}{8} - \frac{5}{8} (- \frac{1}{27}) & \frac{1}{8} - \frac{5}{8} \cdot \frac{7}{27} & 0 - \frac{5}{8} \cdot 0 & 0 - \frac{5}{8} \cdot \frac{4}{27} & \frac{29}{8} - \frac{5}{8} \cdot \frac{211}{27} \\ 0 & 1 & 0 & - \frac{23}{27} & - \frac{1}{27} & 0 & - \frac{5}{54} & - \frac{331}{27} \\ 0 & 0 & 1 & - \frac{1}{27} & \frac{7}{27} & 0 & \frac{4}{27} & \frac{211}{27} \end{array}]$

Schritt 6.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{4}{27} & - \frac{1}{27} & 0 & - \frac{5}{54} & - \frac{34}{27} \\ 0 & 1 & 0 & - \frac{23}{27} & - \frac{1}{27} & 0 & - \frac{5}{54} & - \frac{331}{27} \\ 0 & 0 & 1 & - \frac{1}{27} & \frac{7}{27} & 0 & \frac{4}{27} & \frac{211}{27} \end{array}]$