Lineare Algebra Beispiele

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ - 1 & 1 & 3 & - 4 & - 15 \\ 8 & 7 & - 4 & 36 & 131 \end{array}]$

Schritt 1

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

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1

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

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ - 1 + 1 \cdot 1 & 1 + 1 \cdot 2 & 3 + 1 \cdot 1 & - 4 + 1 \cdot 3 & - 15 + 1 \cdot 10 \\ 8 & 7 & - 4 & 36 & 131 \end{array}]$

Schritt 1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 3 & 4 & - 1 & - 5 \\ 8 & 7 & - 4 & 36 & 131 \end{array}]$

Schritt 2

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 3 & 4 & - 1 & - 5 \\ 8 - 8 \cdot 1 & 7 - 8 \cdot 2 & - 4 - 8 \cdot 1 & 36 - 8 \cdot 3 & 131 - 8 \cdot 10 \end{array}]$

Schritt 2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 3 & 4 & - 1 & - 5 \\ 0 & - 9 & - 12 & 12 & 51 \end{array}]$

Schritt 3

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

Tippen, um mehr Schritte zu sehen ...

Schritt 3.1

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

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ \frac{0}{3} & \frac{3}{3} & \frac{4}{3} & \frac{- 1}{3} & \frac{- 5}{3} \\ 0 & - 9 & - 12 & 12 & 51 \end{array}]$

Schritt 3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 1 & \frac{4}{3} & - \frac{1}{3} & - \frac{5}{3} \\ 0 & - 9 & - 12 & 12 & 51 \end{array}]$

Schritt 4

Perform the row operation $R_{3} = R_{3} + 9 R_{2}$ to make the entry at $3, 2$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 4.1

Perform the row operation $R_{3} = R_{3} + 9 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 1 & \frac{4}{3} & - \frac{1}{3} & - \frac{5}{3} \\ 0 + 9 \cdot 0 & - 9 + 9 \cdot 1 & - 12 + 9 (\frac{4}{3}) & 12 + 9 (- \frac{1}{3}) & 51 + 9 (- \frac{5}{3}) \end{array}]$

Schritt 4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 1 & \frac{4}{3} & - \frac{1}{3} & - \frac{5}{3} \\ 0 & 0 & 0 & 9 & 36 \end{array}]$

Schritt 5

Multiply each element of $R_{3}$ by $\frac{1}{9}$ to make the entry at $3, 4$ a $1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 5.1

Multiply each element of $R_{3}$ by $\frac{1}{9}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 1 & \frac{4}{3} & - \frac{1}{3} & - \frac{5}{3} \\ \frac{0}{9} & \frac{0}{9} & \frac{0}{9} & \frac{9}{9} & \frac{36}{9} \end{array}]$

Schritt 5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 1 & \frac{4}{3} & - \frac{1}{3} & - \frac{5}{3} \\ 0 & 0 & 0 & 1 & 4 \end{array}]$

Schritt 6

Perform the row operation $R_{2} = R_{2} + \frac{1}{3} R_{3}$ to make the entry at $2, 4$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 6.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{3} R_{3}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 + \frac{1}{3} \cdot 0 & 1 + \frac{1}{3} \cdot 0 & \frac{4}{3} + \frac{1}{3} \cdot 0 & - \frac{1}{3} + \frac{1}{3} \cdot 1 & - \frac{5}{3} + \frac{1}{3} \cdot 4 \\ 0 & 0 & 0 & 1 & 4 \end{array}]$

Schritt 6.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 3 & 10 \\ 0 & 1 & \frac{4}{3} & 0 & - \frac{1}{3} \\ 0 & 0 & 0 & 1 & 4 \end{array}]$

Schritt 7

Perform the row operation $R_{1} = R_{1} - 3 R_{3}$ to make the entry at $1, 4$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 7.1

Perform the row operation $R_{1} = R_{1} - 3 R_{3}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 3 \cdot 0 & 2 - 3 \cdot 0 & 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & 10 - 3 \cdot 4 \\ 0 & 1 & \frac{4}{3} & 0 & - \frac{1}{3} \\ 0 & 0 & 0 & 1 & 4 \end{array}]$

Schritt 7.2

Vereinfache $R_{1}$ .

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

Schritt 8

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

Tippen, um mehr Schritte zu sehen ...

Schritt 8.1

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

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

Schritt 8.2

Vereinfache $R_{1}$ .

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