Lineare Algebra Beispiele

$[\begin{array}{c} 2 & 2 & 2 & 0 \\ - 2 & 5 & 2 & 1 \\ 8 & 1 & 4 & - 1 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{2}{2} & \frac{2}{2} & \frac{0}{2} \\ - 2 & 5 & 2 & 1 \\ 8 & 1 & 4 & - 1 \end{array}]$

Schritt 1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ - 2 & 5 & 2 & 1 \\ 8 & 1 & 4 & - 1 \end{array}]$

Schritt 2

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 2.1

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

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ - 2 + 2 \cdot 1 & 5 + 2 \cdot 1 & 2 + 2 \cdot 1 & 1 + 2 \cdot 0 \\ 8 & 1 & 4 & - 1 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 7 & 4 & 1 \\ 8 & 1 & 4 & - 1 \end{array}]$

Schritt 3

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 3.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 7 & 4 & 1 \\ 8 - 8 \cdot 1 & 1 - 8 \cdot 1 & 4 - 8 \cdot 1 & - 1 - 8 \cdot 0 \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 7 & 4 & 1 \\ 0 & - 7 & - 4 & - 1 \end{array}]$

Schritt 4

Multiply each element of $R_{2}$ by $\frac{1}{7}$ to make the entry at $2, 2$ a $1$ .

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Schritt 4.1

Multiply each element of $R_{2}$ by $\frac{1}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ \frac{0}{7} & \frac{7}{7} & \frac{4}{7} & \frac{1}{7} \\ 0 & - 7 & - 4 & - 1 \end{array}]$

Schritt 4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 1 & \frac{4}{7} & \frac{1}{7} \\ 0 & - 7 & - 4 & - 1 \end{array}]$

Schritt 5

Perform the row operation $R_{3} = R_{3} + 7 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 5.1

Perform the row operation $R_{3} = R_{3} + 7 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 1 & \frac{4}{7} & \frac{1}{7} \\ 0 + 7 \cdot 0 & - 7 + 7 \cdot 1 & - 4 + 7 (\frac{4}{7}) & - 1 + 7 (\frac{1}{7}) \end{array}]$

Schritt 5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 1 & \frac{4}{7} & \frac{1}{7} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 6

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

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Schritt 6.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 1 - \frac{4}{7} & 0 - \frac{1}{7} \\ 0 & 1 & \frac{4}{7} & \frac{1}{7} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 6.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{3}{7} & - \frac{1}{7} \\ 0 & 1 & \frac{4}{7} & \frac{1}{7} \\ 0 & 0 & 0 & 0 \end{array}]$