Finite Mathematik Beispiele

$[\begin{array}{c} 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 8 \end{array}]$

Schritt 1

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 1 & 8 \\ 0 & 0 & 0 & 2 \end{array}]$

Schritt 2

Multiply each element of $R_{4}$ by $\frac{1}{2}$ to make the entry at $4, 4$ a $1$ .

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Schritt 2.1

Multiply each element of $R_{4}$ by $\frac{1}{2}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 1 & 8 \\ \frac{0}{2} & \frac{0}{2} & \frac{0}{2} & \frac{2}{2} \end{array}]$

Schritt 2.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 1 & 8 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 3

Perform the row operation $R_{3} = R_{3} - 8 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Schritt 3.1

Perform the row operation $R_{3} = R_{3} - 8 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 6 \\ 0 - 8 \cdot 0 & 0 - 8 \cdot 0 & 1 - 8 \cdot 0 & 8 - 8 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 4

Perform the row operation $R_{2} = R_{2} - 6 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Schritt 4.1

Perform the row operation $R_{2} = R_{2} - 6 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 9 \\ 0 - 6 \cdot 0 & 1 - 6 \cdot 0 & 0 - 6 \cdot 0 & 6 - 6 \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 5

Perform the row operation $R_{1} = R_{1} - 9 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Schritt 5.1

Perform the row operation $R_{1} = R_{1} - 9 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 9 \cdot 0 & 0 - 9 \cdot 0 & 0 - 9 \cdot 0 & 9 - 9 \cdot 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 5.2

Vereinfache $R_{1}$ .

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