Algebra Beispiele

$[\begin{array}{c} 3 & 12 & - 14 & 5 \\ 2 & 9 & - 11 & 1 \\ 1 & 3 & - 3 & 4 \end{array}]$

Schritt 1

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

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Schritt 1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{12}{3} & \frac{- 14}{3} & \frac{5}{3} \\ 2 & 9 & - 11 & 1 \\ 1 & 3 & - 3 & 4 \end{array}]$

Schritt 1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 4 & - \frac{14}{3} & \frac{5}{3} \\ 2 & 9 & - 11 & 1 \\ 1 & 3 & - 3 & 4 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 4 & - \frac{14}{3} & \frac{5}{3} \\ 2 - 2 \cdot 1 & 9 - 2 \cdot 4 & - 11 - 2 (- \frac{14}{3}) & 1 - 2 (\frac{5}{3}) \\ 1 & 3 & - 3 & 4 \end{array}]$

Schritt 2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 4 & - \frac{14}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{5}{3} & - \frac{7}{3} \\ 1 & 3 & - 3 & 4 \end{array}]$

Schritt 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 3.1

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

$[\begin{array}{c} 1 & 4 & - \frac{14}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{5}{3} & - \frac{7}{3} \\ 1 - 1 & 3 - 4 & - 3 + \frac{14}{3} & 4 - \frac{5}{3} \end{array}]$

Schritt 3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 4 & - \frac{14}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{5}{3} & - \frac{7}{3} \\ 0 & - 1 & \frac{5}{3} & \frac{7}{3} \end{array}]$

Schritt 4

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

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Schritt 4.1

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

$[\begin{array}{c} 1 & 4 & - \frac{14}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{5}{3} & - \frac{7}{3} \\ 0 + 0 & - 1 + 1 \cdot 1 & \frac{5}{3} - \frac{5}{3} & \frac{7}{3} - \frac{7}{3} \end{array}]$

Schritt 4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 4 & - \frac{14}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{5}{3} & - \frac{7}{3} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 5

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

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Schritt 5.1

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

$[\begin{array}{c} 1 - 4 \cdot 0 & 4 - 4 \cdot 1 & - \frac{14}{3} - 4 (- \frac{5}{3}) & \frac{5}{3} - 4 (- \frac{7}{3}) \\ 0 & 1 & - \frac{5}{3} & - \frac{7}{3} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 5.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 2 & 11 \\ 0 & 1 & - \frac{5}{3} & - \frac{7}{3} \\ 0 & 0 & 0 & 0 \end{array}]$