Finite Mathematik Beispiele

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 4 & 10 & 1 & 0 & 1 & 0 & 10 \\ - 4 & 40 & 1 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

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

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Schritt 1.1.1

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

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 4 - 4 \cdot 1 & 10 - 4 \cdot 2 & 1 - 4 \cdot 3 & 0 - 4 \cdot 1 & 1 - 4 \cdot 0 & 0 - 4 \cdot 0 & 10 - 4 \cdot 8 \\ - 4 & 40 & 1 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 2 & - 11 & - 4 & 1 & 0 & - 22 \\ - 4 & 40 & 1 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 1.2

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 1.2.1

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

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 2 & - 11 & - 4 & 1 & 0 & - 22 \\ - 4 + 4 \cdot 1 & 40 + 4 \cdot 2 & 1 + 4 \cdot 3 & 0 + 4 \cdot 1 & 0 + 4 \cdot 0 & 1 + 4 \cdot 0 & 0 + 4 \cdot 8 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 2 & - 11 & - 4 & 1 & 0 & - 22 \\ 0 & 48 & 13 & 4 & 0 & 1 & 32 \end{array}]$

Schritt 1.3

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

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Schritt 1.3.1

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

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ \frac{0}{2} & \frac{2}{2} & \frac{- 11}{2} & \frac{- 4}{2} & \frac{1}{2} & \frac{0}{2} & \frac{- 22}{2} \\ 0 & 48 & 13 & 4 & 0 & 1 & 32 \end{array}]$

Schritt 1.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 1 & - \frac{11}{2} & - 2 & \frac{1}{2} & 0 & - 11 \\ 0 & 48 & 13 & 4 & 0 & 1 & 32 \end{array}]$

Schritt 1.4

Perform the row operation $R_{3} = R_{3} - 48 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 1.4.1

Perform the row operation $R_{3} = R_{3} - 48 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 1 & - \frac{11}{2} & - 2 & \frac{1}{2} & 0 & - 11 \\ 0 - 48 \cdot 0 & 48 - 48 \cdot 1 & 13 - 48 (- \frac{11}{2}) & 4 - 48 \cdot - 2 & 0 - 48 (\frac{1}{2}) & 1 - 48 \cdot 0 & 32 - 48 \cdot - 11 \end{array}]$

Schritt 1.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 1 & - \frac{11}{2} & - 2 & \frac{1}{2} & 0 & - 11 \\ 0 & 0 & 277 & 100 & - 24 & 1 & 560 \end{array}]$

Schritt 1.5

Multiply each element of $R_{3}$ by $\frac{1}{277}$ to make the entry at $3, 3$ a $1$ .

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Schritt 1.5.1

Multiply each element of $R_{3}$ by $\frac{1}{277}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 1 & - \frac{11}{2} & - 2 & \frac{1}{2} & 0 & - 11 \\ \frac{0}{277} & \frac{0}{277} & \frac{277}{277} & \frac{100}{277} & \frac{- 24}{277} & \frac{1}{277} & \frac{560}{277} \end{array}]$

Schritt 1.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 1 & - \frac{11}{2} & - 2 & \frac{1}{2} & 0 & - 11 \\ 0 & 0 & 1 & \frac{100}{277} & - \frac{24}{277} & \frac{1}{277} & \frac{560}{277} \end{array}]$

Schritt 1.6

Perform the row operation $R_{2} = R_{2} + \frac{11}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 1.6.1

Perform the row operation $R_{2} = R_{2} + \frac{11}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 + \frac{11}{2} \cdot 0 & 1 + \frac{11}{2} \cdot 0 & - \frac{11}{2} + \frac{11}{2} \cdot 1 & - 2 + \frac{11}{2} \cdot \frac{100}{277} & \frac{1}{2} + \frac{11}{2} (- \frac{24}{277}) & 0 + \frac{11}{2} \cdot \frac{1}{277} & - 11 + \frac{11}{2} \cdot \frac{560}{277} \\ 0 & 0 & 1 & \frac{100}{277} & - \frac{24}{277} & \frac{1}{277} & \frac{560}{277} \end{array}]$

Schritt 1.6.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 & 8 \\ 0 & 1 & 0 & - \frac{4}{277} & \frac{13}{554} & \frac{11}{554} & \frac{33}{277} \\ 0 & 0 & 1 & \frac{100}{277} & - \frac{24}{277} & \frac{1}{277} & \frac{560}{277} \end{array}]$

Schritt 1.7

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

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Schritt 1.7.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & 1 - 3 (\frac{100}{277}) & 0 - 3 (- \frac{24}{277}) & 0 - 3 (\frac{1}{277}) & 8 - 3 (\frac{560}{277}) \\ 0 & 1 & 0 & - \frac{4}{277} & \frac{13}{554} & \frac{11}{554} & \frac{33}{277} \\ 0 & 0 & 1 & \frac{100}{277} & - \frac{24}{277} & \frac{1}{277} & \frac{560}{277} \end{array}]$

Schritt 1.7.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & - \frac{23}{277} & \frac{72}{277} & - \frac{3}{277} & \frac{536}{277} \\ 0 & 1 & 0 & - \frac{4}{277} & \frac{13}{554} & \frac{11}{554} & \frac{33}{277} \\ 0 & 0 & 1 & \frac{100}{277} & - \frac{24}{277} & \frac{1}{277} & \frac{560}{277} \end{array}]$

Schritt 1.8

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

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Schritt 1.8.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & - \frac{23}{277} - 2 (- \frac{4}{277}) & \frac{72}{277} - 2 (\frac{13}{554}) & - \frac{3}{277} - 2 (\frac{11}{554}) & \frac{536}{277} - 2 (\frac{33}{277}) \\ 0 & 1 & 0 & - \frac{4}{277} & \frac{13}{554} & \frac{11}{554} & \frac{33}{277} \\ 0 & 0 & 1 & \frac{100}{277} & - \frac{24}{277} & \frac{1}{277} & \frac{560}{277} \end{array}]$

Schritt 1.8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{15}{277} & \frac{59}{277} & - \frac{14}{277} & \frac{470}{277} \\ 0 & 1 & 0 & - \frac{4}{277} & \frac{13}{554} & \frac{11}{554} & \frac{33}{277} \\ 0 & 0 & 1 & \frac{100}{277} & - \frac{24}{277} & \frac{1}{277} & \frac{560}{277} \end{array}]$

Schritt 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22},$ and $a_{33}$

Pivot Columns: $1, 2,$ and $3$