Finite Mathematik Beispiele

$[\begin{array}{c} 10 & 10 & 1 & 0 & 0 & 7500 \\ 20 & 30 & 0 & 1 & 0 & 19500 \\ - 7 & - 8.5 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1

Multiply each element of $R_{1}$ by $\frac{1}{10}$ to make the entry at $1, 1$ a $1$ .

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Schritt 1.1

Multiply each element of $R_{1}$ by $\frac{1}{10}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{10}{10} & \frac{10}{10} & \frac{1}{10} & \frac{0}{10} & \frac{0}{10} & \frac{7500}{10} \\ 20 & 30 & 0 & 1 & 0 & 19500 \\ - 7 & - 8.5 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 20 & 30 & 0 & 1 & 0 & 19500 \\ - 7 & - 8.5 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2

Perform the row operation $R_{2} = R_{2} - 20 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.1

Perform the row operation $R_{2} = R_{2} - 20 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 20 - 20 \cdot 1 & 30 - 20 \cdot 1 & 0 - 20 (\frac{1}{10}) & 1 - 20 \cdot 0 & 0 - 20 \cdot 0 & 19500 - 20 \cdot 750 \\ - 7 & - 8.5 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 10 & - 2 & 1 & 0 & 4500 \\ - 7 & - 8.5 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 3

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

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Schritt 3.1

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

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 10 & - 2 & 1 & 0 & 4500 \\ - 7 + 7 \cdot 1 & - 8.5 + 7 \cdot 1 & 0 + 7 (\frac{1}{10}) & 0 + 7 \cdot 0 & 1 + 7 \cdot 0 & 0 + 7 \cdot 750 \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 10 & - 2 & 1 & 0 & 4500 \\ 0 & - 1.5 & \frac{7}{10} & 0 & 1 & 5250 \end{array}]$

Schritt 4

Multiply each element of $R_{2}$ by $\frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

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Schritt 4.1

Multiply each element of $R_{2}$ by $\frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ \frac{0}{10} & \frac{10}{10} & \frac{- 2}{10} & \frac{1}{10} & \frac{0}{10} & \frac{4500}{10} \\ 0 & - 1.5 & \frac{7}{10} & 0 & 1 & 5250 \end{array}]$

Schritt 4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 1 & - \frac{1}{5} & \frac{1}{10} & 0 & 450 \\ 0 & - 1.5 & \frac{7}{10} & 0 & 1 & 5250 \end{array}]$

Schritt 5

Perform the row operation $R_{3} = R_{3} + 1.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 5.1

Perform the row operation $R_{3} = R_{3} + 1.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 1 & - \frac{1}{5} & \frac{1}{10} & 0 & 450 \\ 0 + 1.5 \cdot 0 & - 1.5 + 1.5 \cdot 1 & \frac{7}{10} + 1.5 (- \frac{1}{5}) & 0 + 1.5 (\frac{1}{10}) & 1 + 1.5 \cdot 0 & 5250 + 1.5 \cdot 450 \end{array}]$

Schritt 5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 1 & - \frac{1}{5} & \frac{1}{10} & 0 & 450 \\ 0 & 0 & \frac{2}{5} & 0.15 & 1 & 5925 \end{array}]$

Schritt 6

Multiply each element of $R_{3}$ by $\frac{5}{2}$ to make the entry at $3, 3$ a $1$ .

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Schritt 6.1

Multiply each element of $R_{3}$ by $\frac{5}{2}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 1 & - \frac{1}{5} & \frac{1}{10} & 0 & 450 \\ \frac{5}{2} \cdot 0 & \frac{5}{2} \cdot 0 & \frac{5}{2} \cdot \frac{2}{5} & \frac{5}{2} \cdot 0.15 & \frac{5}{2} \cdot 1 & \frac{5}{2} \cdot 5925 \end{array}]$

Schritt 6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 1 & - \frac{1}{5} & \frac{1}{10} & 0 & 450 \\ 0 & 0 & 1 & 0.375 & \frac{5}{2} & \frac{29625}{2} \end{array}]$

Schritt 7

Perform the row operation $R_{2} = R_{2} + \frac{1}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 7.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 + \frac{1}{5} \cdot 0 & 1 + \frac{1}{5} \cdot 0 & - \frac{1}{5} + \frac{1}{5} \cdot 1 & \frac{1}{10} + \frac{1}{5} \cdot 0.375 & 0 + \frac{1}{5} \cdot \frac{5}{2} & 450 + \frac{1}{5} \cdot \frac{29625}{2} \\ 0 & 0 & 1 & 0.375 & \frac{5}{2} & \frac{29625}{2} \end{array}]$

Schritt 7.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & \frac{1}{10} & 0 & 0 & 750 \\ 0 & 1 & 0 & 0.175 & \frac{1}{2} & \frac{6825}{2} \\ 0 & 0 & 1 & 0.375 & \frac{5}{2} & \frac{29625}{2} \end{array}]$

Schritt 8

Perform the row operation $R_{1} = R_{1} - \frac{1}{10} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 8.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{10} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{10} \cdot 0 & 1 - \frac{1}{10} \cdot 0 & \frac{1}{10} - \frac{1}{10} \cdot 1 & 0 - \frac{1}{10} \cdot 0.375 & 0 - \frac{1}{10} \cdot \frac{5}{2} & 750 - \frac{1}{10} \cdot \frac{29625}{2} \\ 0 & 1 & 0 & 0.175 & \frac{1}{2} & \frac{6825}{2} \\ 0 & 0 & 1 & 0.375 & \frac{5}{2} & \frac{29625}{2} \end{array}]$

Schritt 8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & - 0.0375 & - \frac{1}{4} & - \frac{2925}{4} \\ 0 & 1 & 0 & 0.175 & \frac{1}{2} & \frac{6825}{2} \\ 0 & 0 & 1 & 0.375 & \frac{5}{2} & \frac{29625}{2} \end{array}]$

Schritt 9

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

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Schritt 9.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & - 0.0375 - 0.175 & - \frac{1}{4} - \frac{1}{2} & - \frac{2925}{4} - \frac{6825}{2} \\ 0 & 1 & 0 & 0.175 & \frac{1}{2} & \frac{6825}{2} \\ 0 & 0 & 1 & 0.375 & \frac{5}{2} & \frac{29625}{2} \end{array}]$

Schritt 9.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 0.2125 & - \frac{3}{4} & - \frac{16575}{4} \\ 0 & 1 & 0 & 0.175 & \frac{1}{2} & \frac{6825}{2} \\ 0 & 0 & 1 & 0.375 & \frac{5}{2} & \frac{29625}{2} \end{array}]$