Finite Mathematik Beispiele

$[\begin{array}{c} 2 & 1 & 1 & 350 \\ 0 & 1 & 3 & 350 \\ 2 & 2 & 1 & 400 \end{array}]$

Schritt 1

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

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Schritt 1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{1}{2} & \frac{1}{2} & \frac{350}{2} \\ 0 & 1 & 3 & 350 \\ 2 & 2 & 1 & 400 \end{array}]$

Schritt 1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 & 1 & 3 & 350 \\ 2 & 2 & 1 & 400 \end{array}]$

Schritt 2

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 2.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 & 1 & 3 & 350 \\ 2 - 2 \cdot 1 & 2 - 2 (\frac{1}{2}) & 1 - 2 (\frac{1}{2}) & 400 - 2 \cdot 175 \end{array}]$

Schritt 2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 & 1 & 3 & 350 \\ 0 & 1 & 0 & 50 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 & 1 & 3 & 350 \\ 0 - 0 & 1 - 1 & 0 - 3 & 50 - 350 \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 & 1 & 3 & 350 \\ 0 & 0 & - 3 & - 300 \end{array}]$

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 $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 & 1 & 3 & 350 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 300 \end{array}]$

Schritt 4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 & 1 & 3 & 350 \\ 0 & 0 & 1 & 100 \end{array}]$

Schritt 5

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

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Schritt 5.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 - 3 \cdot 0 & 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & 350 - 3 \cdot 100 \\ 0 & 0 & 1 & 100 \end{array}]$

Schritt 5.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 175 \\ 0 & 1 & 0 & 50 \\ 0 & 0 & 1 & 100 \end{array}]$

Schritt 6

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

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Schritt 6.1

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

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 175 - \frac{1}{2} \cdot 100 \\ 0 & 1 & 0 & 50 \\ 0 & 0 & 1 & 100 \end{array}]$

Schritt 6.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 125 \\ 0 & 1 & 0 & 50 \\ 0 & 0 & 1 & 100 \end{array}]$

Schritt 7

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

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Schritt 7.1

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

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 0 - \frac{1}{2} \cdot 0 & 125 - \frac{1}{2} \cdot 50 \\ 0 & 1 & 0 & 50 \\ 0 & 0 & 1 & 100 \end{array}]$

Schritt 7.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 100 \\ 0 & 1 & 0 & 50 \\ 0 & 0 & 1 & 100 \end{array}]$