Lineare Algebra Beispiele

$[\begin{array}{c} 2 & 5 & 1 \\ - 1 & 0 & 1 \\ 3 & 4 & - 1 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{5}{2} & \frac{1}{2} \\ - 1 & 0 & 1 \\ 3 & 4 & - 1 \end{array}]$

Schritt 1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{1}{2} \\ - 1 & 0 & 1 \\ 3 & 4 & - 1 \end{array}]$

Schritt 2

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

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Schritt 2.1

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

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{1}{2} \\ - 1 + 1 \cdot 1 & 0 + \frac{5}{2} & 1 + \frac{1}{2} \\ 3 & 4 & - 1 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{1}{2} \\ 0 & \frac{5}{2} & \frac{3}{2} \\ 3 & 4 & - 1 \end{array}]$

Schritt 3

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

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Schritt 3.1

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

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{1}{2} \\ 0 & \frac{5}{2} & \frac{3}{2} \\ 3 - 3 \cdot 1 & 4 - 3 (\frac{5}{2}) & - 1 - 3 (\frac{1}{2}) \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

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

Schritt 4

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

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Schritt 4.1

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

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

Schritt 4.2

Vereinfache $R_{2}$ .

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

Schritt 5

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

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Schritt 5.1

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

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

Schritt 5.2

Vereinfache $R_{3}$ .

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

Schritt 6

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

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Schritt 6.1

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

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

Schritt 6.2

Vereinfache $R_{3}$ .

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

Schritt 7

Perform the row operation $R_{2} = R_{2} - \frac{3}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 7.1

Perform the row operation $R_{2} = R_{2} - \frac{3}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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

Schritt 7.2

Vereinfache $R_{2}$ .

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

Schritt 8

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 8.1

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

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

Schritt 8.2

Vereinfache $R_{1}$ .

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

Schritt 9

Perform the row operation $R_{1} = R_{1} - \frac{5}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 9.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Schritt 9.2

Vereinfache $R_{1}$ .

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