Álgebra linear Exemplos

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 2 & - 6 & 6 & 10 & - 9 \\ 3 & - 9 & 1 & 7 & - 5 \\ - 2 & 6 & - 14 & - 18 & 0 \end{array}]$

Etapa 1

Encontre a forma escalonada reduzida por linhas.

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Etapa 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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Etapa 1.1.1

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

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 2 - 2 \cdot 1 & - 6 - 2 \cdot - 3 & 6 - 2 \cdot 2 & 10 - 2 \cdot 4 & - 9 - 2 \cdot - 5 \\ 3 & - 9 & 1 & 7 & - 5 \\ - 2 & 6 & - 14 & - 18 & 0 \end{array}]$

Etapa 1.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 2 & 2 & 1 \\ 3 & - 9 & 1 & 7 & - 5 \\ - 2 & 6 & - 14 & - 18 & 0 \end{array}]$

Etapa 1.2

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

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Etapa 1.2.1

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

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 2 & 2 & 1 \\ 3 - 3 \cdot 1 & - 9 - 3 \cdot - 3 & 1 - 3 \cdot 2 & 7 - 3 \cdot 4 & - 5 - 3 \cdot - 5 \\ - 2 & 6 & - 14 & - 18 & 0 \end{array}]$

Etapa 1.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 2 & 2 & 1 \\ 0 & 0 & - 5 & - 5 & 10 \\ - 2 & 6 & - 14 & - 18 & 0 \end{array}]$

Etapa 1.3

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

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Etapa 1.3.1

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

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 2 & 2 & 1 \\ 0 & 0 & - 5 & - 5 & 10 \\ - 2 + 2 \cdot 1 & 6 + 2 \cdot - 3 & - 14 + 2 \cdot 2 & - 18 + 2 \cdot 4 & 0 + 2 \cdot - 5 \end{array}]$

Etapa 1.3.2

Simplifique $R_{4}$ .

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 2 & 2 & 1 \\ 0 & 0 & - 5 & - 5 & 10 \\ 0 & 0 & - 10 & - 10 & - 10 \end{array}]$

Etapa 1.4

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

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Etapa 1.4.1

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

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ \frac{0}{2} & \frac{0}{2} & \frac{2}{2} & \frac{2}{2} & \frac{1}{2} \\ 0 & 0 & - 5 & - 5 & 10 \\ 0 & 0 & - 10 & - 10 & - 10 \end{array}]$

Etapa 1.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 1 & 1 & \frac{1}{2} \\ 0 & 0 & - 5 & - 5 & 10 \\ 0 & 0 & - 10 & - 10 & - 10 \end{array}]$

Etapa 1.5

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

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Etapa 1.5.1

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

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 1 & 1 & \frac{1}{2} \\ 0 + 5 \cdot 0 & 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - 5 + 5 \cdot 1 & 10 + 5 (\frac{1}{2}) \\ 0 & 0 & - 10 & - 10 & - 10 \end{array}]$

Etapa 1.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 1 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & \frac{25}{2} \\ 0 & 0 & - 10 & - 10 & - 10 \end{array}]$

Etapa 1.6

Perform the row operation $R_{4} = R_{4} + 10 R_{2}$ to make the entry at $4, 3$ a $0$ .

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Etapa 1.6.1

Perform the row operation $R_{4} = R_{4} + 10 R_{2}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 1 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & \frac{25}{2} \\ 0 + 10 \cdot 0 & 0 + 10 \cdot 0 & - 10 + 10 \cdot 1 & - 10 + 10 \cdot 1 & - 10 + 10 (\frac{1}{2}) \end{array}]$

Etapa 1.6.2

Simplifique $R_{4}$ .

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 1 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & \frac{25}{2} \\ 0 & 0 & 0 & 0 & - 5 \end{array}]$

Etapa 1.7

Multiply each element of $R_{3}$ by $\frac{2}{25}$ to make the entry at $3, 5$ a $1$ .

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Etapa 1.7.1

Multiply each element of $R_{3}$ by $\frac{2}{25}$ to make the entry at $3, 5$ a $1$ .

$[\begin{array}{c} 1 & - 3 & 2 & 4 & - 5 \\ 0 & 0 & 1 & 1 & \frac{1}{2} \\ \frac{2}{25} \cdot 0 & \frac{2}{25} \cdot 0 & \frac{2}{25} \cdot 0 & \frac{2}{25} \cdot 0 & \frac{2}{25} \cdot \frac{25}{2} \\ 0 & 0 & 0 & 0 & - 5 \end{array}]$

Etapa 1.7.2

Simplifique $R_{3}$ .

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

Etapa 1.8

Perform the row operation $R_{4} = R_{4} + 5 R_{3}$ to make the entry at $4, 5$ a $0$ .

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Etapa 1.8.1

Perform the row operation $R_{4} = R_{4} + 5 R_{3}$ to make the entry at $4, 5$ a $0$ .

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

Etapa 1.8.2

Simplifique $R_{4}$ .

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

Etapa 1.9

Perform the row operation $R_{2} = R_{2} - \frac{1}{2} R_{3}$ to make the entry at $2, 5$ a $0$ .

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Etapa 1.9.1

Perform the row operation $R_{2} = R_{2} - \frac{1}{2} R_{3}$ to make the entry at $2, 5$ a $0$ .

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

Etapa 1.9.2

Simplifique $R_{2}$ .

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

Etapa 1.10

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

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Etapa 1.10.1

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

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

Etapa 1.10.2

Simplifique $R_{1}$ .

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

Etapa 1.11

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

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Etapa 1.11.1

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

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

Etapa 1.11.2

Simplifique $R_{1}$ .

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

Etapa 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: $a_{11}, a_{23},$ and $a_{35}$

Pivot Columns: $1, 3,$ and $5$