Lineare Algebra Beispiele

$x + 2 y + x = 4$ , $x - y = 1$ , $x + 3 y = 0$

Schritt 1

Addiere $x$ und $x$ .

$2 x + 2 y = 4, x - y = 1, x + 3 y = 0$

Schritt 2

Schreibe das Gleichungssystem in Matrixform.

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

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

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

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Schritt 3.1.1

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

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

Schritt 3.1.2

Vereinfache $R_{1}$ .

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

Schritt 3.2

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

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Schritt 3.2.1

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

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

Schritt 3.2.2

Vereinfache $R_{2}$ .

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

Schritt 3.3

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

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Schritt 3.3.1

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

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

Schritt 3.3.2

Vereinfache $R_{3}$ .

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

Schritt 3.4

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

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Schritt 3.4.1

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

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

Schritt 3.4.2

Vereinfache $R_{2}$ .

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

Schritt 3.5

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

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Schritt 3.5.1

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

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

Schritt 3.5.2

Vereinfache $R_{3}$ .

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

Schritt 3.6

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

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Schritt 3.6.1

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

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

Schritt 3.6.2

Vereinfache $R_{3}$ .

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

Schritt 3.7

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

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Schritt 3.7.1

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

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

Schritt 3.7.2

Vereinfache $R_{2}$ .

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

Schritt 3.8

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

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Schritt 3.8.1

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

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

Schritt 3.8.2

Vereinfache $R_{1}$ .

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

Schritt 3.9

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

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Schritt 3.9.1

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

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

Schritt 3.9.2

Vereinfache $R_{1}$ .

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

Schritt 4

Verwende die Ergebnismatrix, um die endgültigen Lösungen für das Gleichungssystem anzugeben.

$x = 0$

$y = 0$

$0 = 1$

Schritt 5

Da $0 \neq 1$ , gibt es keine Lösungen.

Keine Lösung