Algebra Beispiele

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 2 & 18 & 18 & 17 \\ - 3 & - 30 & - 4 & - 29 \\ 1 & 10 & 2 & 10 \end{array}]$

Schritt 1

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 1.1

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

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 2 - 2 \cdot 1 & 18 - 2 \cdot 10 & 18 - 2 \cdot 2 & 17 - 2 \cdot 7 \\ - 3 & - 30 & - 4 & - 29 \\ 1 & 10 & 2 & 10 \end{array}]$

Schritt 1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & - 2 & 14 & 3 \\ - 3 & - 30 & - 4 & - 29 \\ 1 & 10 & 2 & 10 \end{array}]$

Schritt 2

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

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Schritt 2.1

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

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & - 2 & 14 & 3 \\ - 3 + 3 \cdot 1 & - 30 + 3 \cdot 10 & - 4 + 3 \cdot 2 & - 29 + 3 \cdot 7 \\ 1 & 10 & 2 & 10 \end{array}]$

Schritt 2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & - 2 & 14 & 3 \\ 0 & 0 & 2 & - 8 \\ 1 & 10 & 2 & 10 \end{array}]$

Schritt 3

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

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Schritt 3.1

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

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & - 2 & 14 & 3 \\ 0 & 0 & 2 & - 8 \\ 1 - 1 & 10 - 10 & 2 - 2 & 10 - 7 \end{array}]$

Schritt 3.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & - 2 & 14 & 3 \\ 0 & 0 & 2 & - 8 \\ 0 & 0 & 0 & 3 \end{array}]$

Schritt 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 4.1

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

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 3 \\ 0 & 0 & 2 & - 8 \\ 0 & 0 & 0 & 3 \end{array}]$

Schritt 4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & 1 & - 7 & - \frac{3}{2} \\ 0 & 0 & 2 & - 8 \\ 0 & 0 & 0 & 3 \end{array}]$

Schritt 5

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

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Schritt 5.1

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

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & 1 & - 7 & - \frac{3}{2} \\ \frac{0}{2} & \frac{0}{2} & \frac{2}{2} & \frac{- 8}{2} \\ 0 & 0 & 0 & 3 \end{array}]$

Schritt 5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & 1 & - 7 & - \frac{3}{2} \\ 0 & 0 & 1 & - 4 \\ 0 & 0 & 0 & 3 \end{array}]$

Schritt 6

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

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Schritt 6.1

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

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & 1 & - 7 & - \frac{3}{2} \\ 0 & 0 & 1 & - 4 \\ \frac{0}{3} & \frac{0}{3} & \frac{0}{3} & \frac{3}{3} \end{array}]$

Schritt 6.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & 1 & - 7 & - \frac{3}{2} \\ 0 & 0 & 1 & - 4 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 7

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

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Schritt 7.1

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

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & 1 & - 7 & - \frac{3}{2} \\ 0 + 4 \cdot 0 & 0 + 4 \cdot 0 & 1 + 4 \cdot 0 & - 4 + 4 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 7.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & 1 & - 7 & - \frac{3}{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 8

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

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Schritt 8.1

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

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 + \frac{3}{2} \cdot 0 & 1 + \frac{3}{2} \cdot 0 & - 7 + \frac{3}{2} \cdot 0 & - \frac{3}{2} + \frac{3}{2} \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 8.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 10 & 2 & 7 \\ 0 & 1 & - 7 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 9

Perform the row operation $R_{1} = R_{1} - 7 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Schritt 9.1

Perform the row operation $R_{1} = R_{1} - 7 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 7 \cdot 0 & 10 - 7 \cdot 0 & 2 - 7 \cdot 0 & 7 - 7 \cdot 1 \\ 0 & 1 & - 7 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 9.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 10 & 2 & 0 \\ 0 & 1 & - 7 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 10

Perform the row operation $R_{2} = R_{2} + 7 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 10.1

Perform the row operation $R_{2} = R_{2} + 7 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 10 & 2 & 0 \\ 0 + 7 \cdot 0 & 1 + 7 \cdot 0 & - 7 + 7 \cdot 1 & 0 + 7 \cdot 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 10.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 10 & 2 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 11

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 11.1

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

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

Schritt 11.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 10 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 12

Perform the row operation $R_{1} = R_{1} - 10 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 12.1

Perform the row operation $R_{1} = R_{1} - 10 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Schritt 12.2

Vereinfache $R_{1}$ .

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