Algebra lineare Esempi

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

Passaggio 1

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 1.1

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

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Passaggio 1.1.1

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

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

Passaggio 1.1.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 5 & 2 & - 15 \\ 5 & - 8 & 7 & 1 \end{array}]$

Passaggio 1.2

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 1.2.1

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

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 5 & 2 & - 15 \\ 5 - 5 \cdot 1 & - 8 - 5 \cdot - 4 & 7 - 5 \cdot 0 & 1 - 5 \cdot 8 \end{array}]$

Passaggio 1.2.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 5 & 2 & - 15 \\ 0 & 12 & 7 & - 39 \end{array}]$

Passaggio 1.3

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 1.3.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ \frac{0}{5} & \frac{5}{5} & \frac{2}{5} & \frac{- 15}{5} \\ 0 & 12 & 7 & - 39 \end{array}]$

Passaggio 1.3.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 1 & \frac{2}{5} & - 3 \\ 0 & 12 & 7 & - 39 \end{array}]$

Passaggio 1.4

Perform the row operation $R_{3} = R_{3} - 12 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Passaggio 1.4.1

Perform the row operation $R_{3} = R_{3} - 12 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 1 & \frac{2}{5} & - 3 \\ 0 - 12 \cdot 0 & 12 - 12 \cdot 1 & 7 - 12 (\frac{2}{5}) & - 39 - 12 \cdot - 3 \end{array}]$

Passaggio 1.4.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 1 & \frac{2}{5} & - 3 \\ 0 & 0 & \frac{11}{5} & - 3 \end{array}]$

Passaggio 1.5

Multiply each element of $R_{3}$ by $\frac{5}{11}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 1.5.1

Multiply each element of $R_{3}$ by $\frac{5}{11}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 1 & \frac{2}{5} & - 3 \\ \frac{5}{11} \cdot 0 & \frac{5}{11} \cdot 0 & \frac{5}{11} \cdot \frac{11}{5} & \frac{5}{11} \cdot - 3 \end{array}]$

Passaggio 1.5.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 1 & \frac{2}{5} & - 3 \\ 0 & 0 & 1 & - \frac{15}{11} \end{array}]$

Passaggio 1.6

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

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Passaggio 1.6.1

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

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 - \frac{2}{5} \cdot 0 & 1 - \frac{2}{5} \cdot 0 & \frac{2}{5} - \frac{2}{5} \cdot 1 & - 3 - \frac{2}{5} (- \frac{15}{11}) \\ 0 & 0 & 1 & - \frac{15}{11} \end{array}]$

Passaggio 1.6.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 0 & 8 \\ 0 & 1 & 0 & - \frac{27}{11} \\ 0 & 0 & 1 & - \frac{15}{11} \end{array}]$

Passaggio 1.7

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 1.7.1

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

$[\begin{array}{c} 1 + 4 \cdot 0 & - 4 + 4 \cdot 1 & 0 + 4 \cdot 0 & 8 + 4 (- \frac{27}{11}) \\ 0 & 1 & 0 & - \frac{27}{11} \\ 0 & 0 & 1 & - \frac{15}{11} \end{array}]$

Passaggio 1.7.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{20}{11} \\ 0 & 1 & 0 & - \frac{27}{11} \\ 0 & 0 & 1 & - \frac{15}{11} \end{array}]$

Passaggio 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22},$ and $a_{33}$

Pivot Columns: $1, 2,$ and $3$