Beispiele

⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

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

Schritt 1

Schreibe die Matrix als Produkt einer unteren Dreiecksmatrix und einer oberen Dreiecksmatrix.

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 l_{21} & 1 & 0 l_{31} & l_{32} & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} u_{11} & u_{12} & u_{13} 0 & u_{22} & u_{23} 0 & 0 & u_{33} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

Schritt 2

Multipliziere

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 l_{21} & 1 & 0 l_{31} & l_{32} & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} u_{11} & u_{12} & u_{13} 0 & u_{22} & u_{23} 0 & 0 & u_{33} \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}]$ .

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Schritt 2.1

Zwei Matrizen können nur dann multipliziert werden, wenn die Anzahl der Spalten in der ersten Matrix der Anzahl der Zeilen in der zweiten Matrix entspricht. In diesem Fall ist die erste Matrix

3 \times 3

$3 \times 3$ und die zweite Matrix ist

3 \times 3

$3 \times 3$ .

Schritt 2.2

Multipliziere jede Zeile in der ersten Matrix mit jeder Spalte in der zweiten Matrix.

⎡ ⎢ ⎣ \begin{matrix} 1 u_{11} + 0 \cdot 0 + 0 \cdot 0 & 1 u_{12} + 0 u_{22} + 0 \cdot 0 & 1 u_{13} + 0 u_{23} + 0 u_{33} l_{21} u_{11} + 1 \cdot 0 + 0 \cdot 0 & l_{21} u_{12} + 1 u_{22} + 0 \cdot 0 & l_{21} u_{13} + 1 u_{23} + 0 u_{33} l_{31} u_{11} + l_{32} \cdot 0 + 1 \cdot 0 & l_{31} u_{12} + l_{32} u_{22} + 1 \cdot 0 & l_{31} u_{13} + l_{32} u_{23} + 1 u_{33} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 u_{11} + 0 \cdot 0 + 0 \cdot 0 & 1 u_{12} + 0 u_{22} + 0 \cdot 0 & 1 u_{13} + 0 u_{23} + 0 u_{33} \\ l_{21} u_{11} + 1 \cdot 0 + 0 \cdot 0 & l_{21} u_{12} + 1 u_{22} + 0 \cdot 0 & l_{21} u_{13} + 1 u_{23} + 0 u_{33} \\ l_{31} u_{11} + l_{32} \cdot 0 + 1 \cdot 0 & l_{31} u_{12} + l_{32} u_{22} + 1 \cdot 0 & l_{31} u_{13} + l_{32} u_{23} + 1 u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

Schritt 2.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

⎡ ⎢ ⎣ \begin{matrix} u_{11} & u_{12} & u_{13} l_{21} u_{11} & l_{21} u_{12} + u_{22} & l_{21} u_{13} + u_{23} l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + u_{33} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} u_{11} & u_{12} & u_{13} \\ l_{21} u_{11} & l_{21} u_{12} + u_{22} & l_{21} u_{13} + u_{23} \\ l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} u_{11} & u_{12} & u_{13} l_{21} u_{11} & l_{21} u_{12} + u_{22} & l_{21} u_{13} + u_{23} l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + u_{33} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

Schritt 3

Löse.

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Schritt 3.1

Schreibe als lineares Gleichungssystem.

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{11} = 3

$l_{21} u_{11} = 3$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Schritt 3.2

Löse das Gleichungssystem.

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Schritt 3.2.1

Ersetze alle Vorkommen von

u_{11}

$u_{11}$ durch

1

$1$ in jeder Gleichung.

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Schritt 3.2.1.1

Ersetze alle

u_{11}

$u_{11}$ in

l_{21} u_{11} = 3

$l_{21} u_{11} = 3$ durch

1

$1$ .

l_{21} \cdot 1 = 3

$l_{21} \cdot 1 = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Schritt 3.2.1.2

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Schritt 3.2.1.2.1

Mutltipliziere

l_{21}

$l_{21}$ mit

1

$1$ .

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Schritt 3.2.1.3

Ersetze alle

u_{11}

$u_{11}$ in

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$ durch

1

$1$ .

l_{31} \cdot 1 = 1

$l_{31} \cdot 1 = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Schritt 3.2.1.4

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Schritt 3.2.1.4.1

Mutltipliziere

l_{31}

$l_{31}$ mit

1

$1$ .

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Schritt 3.2.2

Ersetze alle Vorkommen von

l_{31}

$l_{31}$ durch

1

$1$ in jeder Gleichung.

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Schritt 3.2.2.1

Ersetze alle

l_{31}

$l_{31}$ in

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$ durch

1

$1$ .

1 \cdot u_{12} + l_{32} u_{22} = - 2

$1 \cdot u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Schritt 3.2.2.2

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Schritt 3.2.2.2.1

Mutltipliziere

u_{12}

$u_{12}$ mit

1

$1$ .

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Schritt 3.2.2.3

Ersetze alle

l_{31}

$l_{31}$ in

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$ durch

1

$1$ .

1 \cdot u_{13} + l_{32} u_{23} + u_{33} = - 5

$1 \cdot u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

Schritt 3.2.2.4

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Schritt 3.2.2.4.1

Mutltipliziere

u_{13}

$u_{13}$ mit

1

$1$ .

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

Schritt 3.2.3

Ersetze alle Vorkommen von

l_{21}

$l_{21}$ durch

3

$3$ in jeder Gleichung.

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Schritt 3.2.3.1

Ersetze alle

l_{21}

$l_{21}$ in

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$ durch

3

$3$ .

3 \cdot u_{12} + u_{22} = 1

$3 \cdot u_{12} + u_{22} = 1$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

Schritt 3.2.3.2

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Schritt 3.2.3.2.1

Mutltipliziere

3

$3$ mit

u_{12}

$u_{12}$ .

3 u_{12} + u_{22} = 1

$3 u_{12} + u_{22} = 1$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

3 u_{12} + u_{22} = 1

$3 u_{12} + u_{22} = 1$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

Schritt 3.2.3.3

Ersetze alle

l_{21}

$l_{21}$ in

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$ durch

3

$3$ .

3 \cdot u_{13} + u_{23} = - 3

$3 \cdot u_{13} + u_{23} = - 3$

3 u_{12} + u_{22} = 1

$3 u_{12} + u_{22} = 1$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.3.4

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Schritt 3.2.3.4.1

Mutltipliziere

3

$3$ mit

u_{13}

$u_{13}$ .

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

3 u_{12} + u_{22} = 1

$3 u_{12} + u_{22} = 1$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

3 u_{12} + u_{22} = 1

$3 u_{12} + u_{22} = 1$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

3 u_{12} + u_{22} = 1

$3 u_{12} + u_{22} = 1$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.4

Ersetze alle Vorkommen von

u_{12}

$u_{12}$ durch

1

$1$ in jeder Gleichung.

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Schritt 3.2.4.1

Ersetze alle

u_{12}

$u_{12}$ in

3 u_{12} + u_{22} = 1

$3 u_{12} + u_{22} = 1$ durch

1

$1$ .

3 (1) + u_{22} = 1

$3 (1) + u_{22} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.4.2

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Schritt 3.2.4.2.1

Mutltipliziere

3

$3$ mit

1

$1$ .

3 + u_{22} = 1

$3 + u_{22} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

3 + u_{22} = 1

$3 + u_{22} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.4.3

Ersetze alle

u_{12}

$u_{12}$ in

u_{12} + l_{32} u_{22} = - 2

$u_{12} + l_{32} u_{22} = - 2$ durch

1

$1$ .

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

3 + u_{22} = 1

$3 + u_{22} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.4.4

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Schritt 3.2.4.4.1

Entferne die Klammern.

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

3 + u_{22} = 1

$3 + u_{22} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

3 + u_{22} = 1

$3 + u_{22} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

3 + u_{22} = 1

$3 + u_{22} = 1$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.5

Ersetze alle Vorkommen von

u_{13}

$u_{13}$ durch

1

$1$ in jeder Gleichung.

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Schritt 3.2.5.1

Bringe alle Terme, die nicht

u_{22}

$u_{22}$ enthalten, auf die rechte Seite der Gleichung.

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Schritt 3.2.5.1.1

Subtrahiere

3

$3$ von beiden Seiten der Gleichung.

u_{22} = 1 - 3

$u_{22} = 1 - 3$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.5.1.2

Subtrahiere

3

$3$ von

1

$1$ .

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.5.2

Ersetze alle

u_{13}

$u_{13}$ in

3 u_{13} + u_{23} = - 3

$3 u_{13} + u_{23} = - 3$ durch

1

$1$ .

3 (1) + u_{23} = - 3

$3 (1) + u_{23} = - 3$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.5.3

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Schritt 3.2.5.3.1

Mutltipliziere

3

$3$ mit

1

$1$ .

3 + u_{23} = - 3

$3 + u_{23} = - 3$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

3 + u_{23} = - 3

$3 + u_{23} = - 3$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.5.4

Ersetze alle

u_{13}

$u_{13}$ in

u_{13} + l_{32} u_{23} + u_{33} = - 5

$u_{13} + l_{32} u_{23} + u_{33} = - 5$ durch

1

$1$ .

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

3 + u_{23} = - 3

$3 + u_{23} = - 3$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.5.5

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Schritt 3.2.5.5.1

Entferne die Klammern.

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

3 + u_{23} = - 3

$3 + u_{23} = - 3$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

3 + u_{23} = - 3

$3 + u_{23} = - 3$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

3 + u_{23} = - 3

$3 + u_{23} = - 3$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.6

Ersetze alle Vorkommen von

u_{22}

$u_{22}$ durch

- 2

$- 2$ in jeder Gleichung.

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Schritt 3.2.6.1

Bringe alle Terme, die nicht

u_{23}

$u_{23}$ enthalten, auf die rechte Seite der Gleichung.

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Schritt 3.2.6.1.1

Subtrahiere

3

$3$ von beiden Seiten der Gleichung.

u_{23} = - 3 - 3

$u_{23} = - 3 - 3$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.6.1.2

Subtrahiere

3

$3$ von

- 3

$- 3$ .

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.6.2

Ersetze alle

u_{22}

$u_{22}$ in

1 + l_{32} u_{22} = - 2

$1 + l_{32} u_{22} = - 2$ durch

- 2

$- 2$ .

1 + l_{32} \cdot - 2 = - 2

$1 + l_{32} \cdot - 2 = - 2$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.6.3

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Schritt 3.2.6.3.1

Bringe

- 2

$- 2$ auf die linke Seite von

l_{32}

$l_{32}$ .

1 - 2 l_{32} = - 2

$1 - 2 l_{32} = - 2$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

1 - 2 l_{32} = - 2

$1 - 2 l_{32} = - 2$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

1 - 2 l_{32} = - 2

$1 - 2 l_{32} = - 2$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.7

Ersetze alle Vorkommen von

u_{23}

$u_{23}$ durch

- 6

$- 6$ in jeder Gleichung.

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Schritt 3.2.7.1

Bringe alle Terme, die nicht

l_{32}

$l_{32}$ enthalten, auf die rechte Seite der Gleichung.

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Schritt 3.2.7.1.1

Subtrahiere

1

$1$ von beiden Seiten der Gleichung.

- 2 l_{32} = - 2 - 1

$- 2 l_{32} = - 2 - 1$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.7.1.2

Subtrahiere

1

$1$ von

- 2

$- 2$ .

- 2 l_{32} = - 3

$- 2 l_{32} = - 3$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

- 2 l_{32} = - 3

$- 2 l_{32} = - 3$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.7.2

Teile jeden Ausdruck in

- 2 l_{32} = - 3

$- 2 l_{32} = - 3$ durch

- 2

$- 2$ und vereinfache.

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Schritt 3.2.7.2.1

Teile jeden Ausdruck in

- 2 l_{32} = - 3

$- 2 l_{32} = - 3$ durch

- 2

$- 2$ .

\frac{- 2 l_{32}}{- 2} = \frac{- 3}{- 2}

$\frac{- 2 l_{32}}{- 2} = \frac{- 3}{- 2}$

u_{23} = - 6

$u_{23} = - 6$

1 + l_{32} u_{23} + u_{33} = - 5

$1 + l_{32} u_{23} + u_{33} = - 5$

u_{22} = - 2

$u_{22} = - 2$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

Schritt 3.2.7.2.2

Vereinfache die linke Seite.

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Schritt 3.2.7.2.2.1

Kürze den gemeinsamen Faktor von

- 2

$- 2$ .

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Schritt 3.2.7.2.2.1.1

Kürze den gemeinsamen Faktor.

$\frac{- 2 l_{32}}{- 2} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.7.2.2.1.2

Dividiere

$l_{32}$ durch

$1$ .

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.7.2.3

Vereinfache die rechte Seite.

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Schritt 3.2.7.2.3.1

Dividieren zweier negativer Zahlen ergibt eine positive Zahl.

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.7.3

Ersetze alle

$u_{23}$ in

$1 + l_{32} u_{23} + u_{33} = - 5$ durch

$- 6$ .

$1 + l_{32} \cdot - 6 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.7.4

Vereinfache die linke Seite.

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Schritt 3.2.7.4.1

Bringe

$- 6$ auf die linke Seite von

$l_{32}$ .

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.8

Ersetze alle Vorkommen von

$l_{32}$ durch

$\frac{3}{2}$ in jeder Gleichung.

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Schritt 3.2.8.1

Ersetze alle

$l_{32}$ in

$1 - 6 l_{32} + u_{33} = - 5$ durch

$\frac{3}{2}$ .

$1 - 6 (\frac{3}{2}) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.8.2

Vereinfache die linke Seite.

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Schritt 3.2.8.2.1

Vereinfache

$1 - 6 (\frac{3}{2}) + u_{33}$ .

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Schritt 3.2.8.2.1.1

Vereinfache jeden Term.

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Schritt 3.2.8.2.1.1.1

Kürze den gemeinsamen Faktor von

$2$ .

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Schritt 3.2.8.2.1.1.1.1

Faktorisiere

$2$ aus

$- 6$ heraus.

$1 + 2 (- 3) (\frac{3}{2}) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.8.2.1.1.1.2

Kürze den gemeinsamen Faktor.

$1 + 2 \cdot (- 3 (\frac{3}{2})) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.8.2.1.1.1.3

Forme den Ausdruck um.

$1 - 3 \cdot 3 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 3 \cdot 3 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.8.2.1.1.2

Mutltipliziere

$- 3$ mit

$3$ .

$1 - 9 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 9 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.8.2.1.2

Subtrahiere

$9$ von

$1$ .

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.9

Bringe alle Terme, die nicht

$u_{33}$ enthalten, auf die rechte Seite der Gleichung.

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Schritt 3.2.9.1

Addiere

$8$ zu beiden Seiten der Gleichung.

$u_{33} = - 5 + 8$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.9.2

Addiere

$- 5$ und

$8$ .

$u_{33} = 3$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$u_{33} = 3$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Schritt 3.2.10

Löse das Gleichungssystem.

$u_{33} = 3 l_{32} = \frac{3}{2} u_{23} = - 6 u_{22} = - 2 l_{31} = 1 l_{21} = 3 u_{11} = 1 u_{12} = 1 u_{13} = 1$

Schritt 3.2.11

Liste alle Lösungen auf.

$u_{33} = 3, l_{32} = \frac{3}{2}, u_{23} = - 6, u_{22} = - 2, l_{31} = 1, l_{21} = 3, u_{11} = 1, u_{12} = 1, u_{13} = 1$

Schritt 4

Setze die gelösten Werte ein.

$[\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}] = [\begin{array}{c} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 1 & \frac{3}{2} & 1 \end{array}] [\begin{array}{c} 1 & 1 & 1 \\ 0 & - 2 & - 6 \\ 0 & 0 & 3 \end{array}]$

Gib DEINE Aufgabe ein