Lineare Algebra Beispiele

$x - y = 9$ , $4 x - 3 y = 1$ , $17 x - 13 y = 10$

Schritt 1

Schreibe das Gleichungssystem in Matrixform.

$[\begin{array}{c} 1 & - 1 & 9 \\ 4 & - 3 & 1 \\ 17 & - 13 & 10 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.1.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 9 \\ 4 - 4 \cdot 1 & - 3 - 4 \cdot - 1 & 1 - 4 \cdot 9 \\ 17 & - 13 & 10 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 9 \\ 0 & 1 & - 35 \\ 17 & - 13 & 10 \end{array}]$

Schritt 2.2

Perform the row operation $R_{3} = R_{3} - 17 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 2.2.1

Perform the row operation $R_{3} = R_{3} - 17 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 9 \\ 0 & 1 & - 35 \\ 17 - 17 \cdot 1 & - 13 - 17 \cdot - 1 & 10 - 17 \cdot 9 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 9 \\ 0 & 1 & - 35 \\ 0 & 4 & - 143 \end{array}]$

Schritt 2.3

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 2.3.1

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 9 \\ 0 & 1 & - 35 \\ 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & - 143 - 4 \cdot - 35 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 9 \\ 0 & 1 & - 35 \\ 0 & 0 & - 3 \end{array}]$

Schritt 2.4

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

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Schritt 2.4.1

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

$[\begin{array}{c} 1 & - 1 & 9 \\ 0 & 1 & - 35 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 \end{array}]$

Schritt 2.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 9 \\ 0 & 1 & - 35 \\ 0 & 0 & 1 \end{array}]$

Schritt 2.5

Perform the row operation $R_{2} = R_{2} + 35 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 2.5.1

Perform the row operation $R_{2} = R_{2} + 35 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 9 \\ 0 + 35 \cdot 0 & 1 + 35 \cdot 0 & - 35 + 35 \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Schritt 2.5.2

Vereinfache $R_{2}$ .

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

Schritt 2.6

Perform the row operation $R_{1} = R_{1} - 9 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 2.6.1

Perform the row operation $R_{1} = R_{1} - 9 R_{3}$ to make the entry at $1, 3$ a $0$ .

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

Schritt 2.6.2

Vereinfache $R_{1}$ .

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

Schritt 2.7

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

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Schritt 2.7.1

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

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

Schritt 2.7.2

Vereinfache $R_{1}$ .

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

Schritt 3

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

$x = 0$

$y = 0$

$0 = 1$

Schritt 4

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

Keine Lösung