Lineare Algebra Beispiele

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 2 & 2 & 9 & 7 \\ 0 & 1 & 5 & - 2 \end{array}]$

Schritt 1

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

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Schritt 1.1

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

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 2 - 2 \cdot 1 & 2 - 2 \cdot 0 & 9 - 2 \cdot - 3 & 7 - 2 \cdot 8 \\ 0 & 1 & 5 & - 2 \end{array}]$

Schritt 1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 0 & 2 & 15 & - 9 \\ 0 & 1 & 5 & - 2 \end{array}]$

Schritt 2

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

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Schritt 2.1

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

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ \frac{0}{2} & \frac{2}{2} & \frac{15}{2} & \frac{- 9}{2} \\ 0 & 1 & 5 & - 2 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 0 & 1 & \frac{15}{2} & - \frac{9}{2} \\ 0 & 1 & 5 & - 2 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 0 & 1 & \frac{15}{2} & - \frac{9}{2} \\ 0 - 0 & 1 - 1 & 5 - \frac{15}{2} & - 2 + \frac{9}{2} \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 0 & 1 & \frac{15}{2} & - \frac{9}{2} \\ 0 & 0 & - \frac{5}{2} & \frac{5}{2} \end{array}]$

Schritt 4

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

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Schritt 4.1

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

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 0 & 1 & \frac{15}{2} & - \frac{9}{2} \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} (- \frac{5}{2}) & - \frac{2}{5} \cdot \frac{5}{2} \end{array}]$

Schritt 4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 0 & 1 & \frac{15}{2} & - \frac{9}{2} \\ 0 & 0 & 1 & - 1 \end{array}]$

Schritt 5

Perform the row operation $R_{2} = R_{2} - \frac{15}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 5.1

Perform the row operation $R_{2} = R_{2} - \frac{15}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 3 & 8 \\ 0 - \frac{15}{2} \cdot 0 & 1 - \frac{15}{2} \cdot 0 & \frac{15}{2} - \frac{15}{2} \cdot 1 & - \frac{9}{2} - \frac{15}{2} \cdot - 1 \\ 0 & 0 & 1 & - 1 \end{array}]$

Schritt 5.2

Vereinfache $R_{2}$ .

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

Schritt 6

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 6.1

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

$[\begin{array}{c} 1 + 3 \cdot 0 & 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 8 + 3 \cdot - 1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & - 1 \end{array}]$

Schritt 6.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & - 1 \end{array}]$