Algebra lineare Esempi

$[\begin{array}{c} 3 & - 2 & 0 \\ 6 & - 4 & - 8 \\ 1 & 12 & - 2 \end{array}]$

Passaggio 1

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

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Passaggio 1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{0}{3} \\ 6 & - 4 & - 8 \\ 1 & 12 & - 2 \end{array}]$

Passaggio 1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 6 & - 4 & - 8 \\ 1 & 12 & - 2 \end{array}]$

Passaggio 2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Passaggio 2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 6 - 6 \cdot 1 & - 4 - 6 (- \frac{2}{3}) & - 8 - 6 \cdot 0 \\ 1 & 12 & - 2 \end{array}]$

Passaggio 2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 0 & - 8 \\ 1 & 12 & - 2 \end{array}]$

Passaggio 3

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

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Passaggio 3.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 0 & - 8 \\ 1 - 1 & 12 + \frac{2}{3} & - 2 - 0 \end{array}]$

Passaggio 3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 0 & - 8 \\ 0 & \frac{38}{3} & - 2 \end{array}]$

Passaggio 4

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & \frac{38}{3} & - 2 \\ 0 & 0 & - 8 \end{array}]$

Passaggio 5

Multiply each element of $R_{2}$ by $\frac{3}{38}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 5.1

Multiply each element of $R_{2}$ by $\frac{3}{38}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ \frac{3}{38} \cdot 0 & \frac{3}{38} \cdot \frac{38}{3} & \frac{3}{38} \cdot - 2 \\ 0 & 0 & - 8 \end{array}]$

Passaggio 5.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & - \frac{3}{19} \\ 0 & 0 & - 8 \end{array}]$

Passaggio 6

Multiply each element of $R_{3}$ by $- \frac{1}{8}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 6.1

Multiply each element of $R_{3}$ by $- \frac{1}{8}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & - \frac{3}{19} \\ - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot - 8 \end{array}]$

Passaggio 6.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & - \frac{3}{19} \\ 0 & 0 & 1 \end{array}]$

Passaggio 7

Perform the row operation $R_{2} = R_{2} + \frac{3}{19} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Passaggio 7.1

Perform the row operation $R_{2} = R_{2} + \frac{3}{19} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 + \frac{3}{19} \cdot 0 & 1 + \frac{3}{19} \cdot 0 & - \frac{3}{19} + \frac{3}{19} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Passaggio 7.2

Semplifica $R_{2}$ .

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

Passaggio 8

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

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Passaggio 8.1

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

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

Passaggio 8.2

Semplifica $R_{1}$ .

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