Algebra lineare Esempi

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ - 1 & 2 & - 4 & 1 \\ 5 & - 6 & 10 & 7 \end{array}]$

Passaggio 1

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

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Passaggio 1.1

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

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ - 1 + 1 \cdot 1 & 2 - 4 & - 4 + 1 \cdot 9 & 1 - 7 \\ 5 & - 6 & 10 & 7 \end{array}]$

Passaggio 1.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ 0 & - 2 & 5 & - 6 \\ 5 & - 6 & 10 & 7 \end{array}]$

Passaggio 2

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 2.1

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

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ 0 & - 2 & 5 & - 6 \\ 5 - 5 \cdot 1 & - 6 - 5 \cdot - 4 & 10 - 5 \cdot 9 & 7 - 5 \cdot - 7 \end{array}]$

Passaggio 2.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ 0 & - 2 & 5 & - 6 \\ 0 & 14 & - 35 & 42 \end{array}]$

Passaggio 3

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

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Passaggio 3.1

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

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 5 & - \frac{1}{2} \cdot - 6 \\ 0 & 14 & - 35 & 42 \end{array}]$

Passaggio 3.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ 0 & 1 & - \frac{5}{2} & 3 \\ 0 & 14 & - 35 & 42 \end{array}]$

Passaggio 4

Perform the row operation $R_{3} = R_{3} - 14 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Passaggio 4.1

Perform the row operation $R_{3} = R_{3} - 14 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ 0 & 1 & - \frac{5}{2} & 3 \\ 0 - 14 \cdot 0 & 14 - 14 \cdot 1 & - 35 - 14 (- \frac{5}{2}) & 42 - 14 \cdot 3 \end{array}]$

Passaggio 4.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 9 & - 7 \\ 0 & 1 & - \frac{5}{2} & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Passaggio 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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Passaggio 5.1

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

$[\begin{array}{c} 1 + 4 \cdot 0 & - 4 + 4 \cdot 1 & 9 + 4 (- \frac{5}{2}) & - 7 + 4 \cdot 3 \\ 0 & 1 & - \frac{5}{2} & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Passaggio 5.2

Semplifica $R_{1}$ .

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