Álgebra linear Exemplos

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

Etapa 1

Encontre a forma escalonada reduzida por linhas.

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Etapa 1.1

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

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Etapa 1.1.1

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

$[\begin{array}{c} 1 & 2 & - 1 \\ - 1 + 1 \cdot 1 & 4 + 1 \cdot 2 & 1 - 1 \\ 0 & 6 & 0 \end{array}]$

Etapa 1.1.2

Simplifique $R_{2}$ .

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

Etapa 1.2

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Etapa 1.2.1

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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

Etapa 1.2.2

Simplifique $R_{2}$ .

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

Etapa 1.3

Perform the row operation $R_{3} = R_{3} - 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 1.3.1

Perform the row operation $R_{3} = R_{3} - 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

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

Etapa 1.3.2

Simplifique $R_{3}$ .

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

Etapa 1.4

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

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Etapa 1.4.1

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

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

Etapa 1.4.2

Simplifique $R_{1}$ .

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

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: $a_{11}$ and $a_{22}$

Pivot Columns: $1$ and $2$

Etapa 3

The rank is the number of pivot columns.

$2$