Algebra lineare Esempi

$[\begin{array}{c} 2 & 4 & - 1 & - 2 & 2 & 6 \\ 1 & 3 & 2 & - 7 & 3 & 9 \\ 5 & 8 & - 7 & 6 & 1 & 4 \end{array}]$

Passaggio 1

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

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Passaggio 1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{4}{2} & \frac{- 1}{2} & \frac{- 2}{2} & \frac{2}{2} & \frac{6}{2} \\ 1 & 3 & 2 & - 7 & 3 & 9 \\ 5 & 8 & - 7 & 6 & 1 & 4 \end{array}]$

Passaggio 1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 2 & - \frac{1}{2} & - 1 & 1 & 3 \\ 1 & 3 & 2 & - 7 & 3 & 9 \\ 5 & 8 & - 7 & 6 & 1 & 4 \end{array}]$

Passaggio 2

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

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Passaggio 2.1

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

$[\begin{array}{c} 1 & 2 & - \frac{1}{2} & - 1 & 1 & 3 \\ 1 - 1 & 3 - 2 & 2 + \frac{1}{2} & - 7 + 1 & 3 - 1 & 9 - 3 \\ 5 & 8 & - 7 & 6 & 1 & 4 \end{array}]$

Passaggio 2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - \frac{1}{2} & - 1 & 1 & 3 \\ 0 & 1 & \frac{5}{2} & - 6 & 2 & 6 \\ 5 & 8 & - 7 & 6 & 1 & 4 \end{array}]$

Passaggio 3

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Passaggio 3.1

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{1}{2} & - 1 & 1 & 3 \\ 0 & 1 & \frac{5}{2} & - 6 & 2 & 6 \\ 5 - 5 \cdot 1 & 8 - 5 \cdot 2 & - 7 - 5 (- \frac{1}{2}) & 6 - 5 \cdot - 1 & 1 - 5 \cdot 1 & 4 - 5 \cdot 3 \end{array}]$

Passaggio 3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 2 & - \frac{1}{2} & - 1 & 1 & 3 \\ 0 & 1 & \frac{5}{2} & - 6 & 2 & 6 \\ 0 & - 2 & - \frac{9}{2} & 11 & - 4 & - 11 \end{array}]$

Passaggio 4

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

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Passaggio 4.1

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

$[\begin{array}{c} 1 & 2 & - \frac{1}{2} & - 1 & 1 & 3 \\ 0 & 1 & \frac{5}{2} & - 6 & 2 & 6 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - \frac{9}{2} + 2 (\frac{5}{2}) & 11 + 2 \cdot - 6 & - 4 + 2 \cdot 2 & - 11 + 2 \cdot 6 \end{array}]$

Passaggio 4.2

Semplifica $R_{3}$ .

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

Passaggio 5

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

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Passaggio 5.1

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

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

Passaggio 5.2

Semplifica $R_{3}$ .

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

Passaggio 6

Perform the row operation $R_{2} = R_{2} - \frac{5}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Passaggio 6.1

Perform the row operation $R_{2} = R_{2} - \frac{5}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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

Passaggio 6.2

Semplifica $R_{2}$ .

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

Passaggio 7

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

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Passaggio 7.1

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

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

Passaggio 7.2

Semplifica $R_{1}$ .

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

Passaggio 8

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

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Passaggio 8.1

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

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

Passaggio 8.2

Semplifica $R_{1}$ .

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