线性代数示例

$B = [\begin{array}{c} 1 & 3 & 6 \\ - 6 & 10 & 12 \\ 3 & - 3 & - 2 \end{array}]$

解题步骤 1

求行简化阶梯形矩阵。

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解题步骤 1.1

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

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解题步骤 1.1.1

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

$[\begin{array}{c} 1 & 3 & 6 \\ - 6 + 6 \cdot 1 & 10 + 6 \cdot 3 & 12 + 6 \cdot 6 \\ 3 & - 3 & - 2 \end{array}]$

解题步骤 1.1.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 28 & 48 \\ 3 & - 3 & - 2 \end{array}]$

解题步骤 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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解题步骤 1.2.1

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

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 28 & 48 \\ 3 - 3 \cdot 1 & - 3 - 3 \cdot 3 & - 2 - 3 \cdot 6 \end{array}]$

解题步骤 1.2.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 28 & 48 \\ 0 & - 12 & - 20 \end{array}]$

解题步骤 1.3

Multiply each element of $R_{2}$ by $\frac{1}{28}$ to make the entry at $2, 2$ a $1$ .

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解题步骤 1.3.1

Multiply each element of $R_{2}$ by $\frac{1}{28}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 3 & 6 \\ \frac{0}{28} & \frac{28}{28} & \frac{48}{28} \\ 0 & - 12 & - 20 \end{array}]$

解题步骤 1.3.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 1 & \frac{12}{7} \\ 0 & - 12 & - 20 \end{array}]$

解题步骤 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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解题步骤 1.4.1

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

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 1 & \frac{12}{7} \\ 0 + 12 \cdot 0 & - 12 + 12 \cdot 1 & - 20 + 12 (\frac{12}{7}) \end{array}]$

解题步骤 1.4.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 1 & \frac{12}{7} \\ 0 & 0 & \frac{4}{7} \end{array}]$

解题步骤 1.5

Multiply each element of $R_{3}$ by $\frac{7}{4}$ to make the entry at $3, 3$ a $1$ .

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解题步骤 1.5.1

Multiply each element of $R_{3}$ by $\frac{7}{4}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 1 & \frac{12}{7} \\ \frac{7}{4} \cdot 0 & \frac{7}{4} \cdot 0 & \frac{7}{4} \cdot \frac{4}{7} \end{array}]$

解题步骤 1.5.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 1 & \frac{12}{7} \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.6

Perform the row operation $R_{2} = R_{2} - \frac{12}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

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解题步骤 1.6.1

Perform the row operation $R_{2} = R_{2} - \frac{12}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 3 & 6 \\ 0 - \frac{12}{7} \cdot 0 & 1 - \frac{12}{7} \cdot 0 & \frac{12}{7} - \frac{12}{7} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.6.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.7

Perform the row operation $R_{1} = R_{1} - 6 R_{3}$ to make the entry at $1, 3$ a $0$ .

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解题步骤 1.7.1

Perform the row operation $R_{1} = R_{1} - 6 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 6 \cdot 0 & 3 - 6 \cdot 0 & 6 - 6 \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.7.2

化简 $R_{1}$ 。

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

解题步骤 1.8

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

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解题步骤 1.8.1

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

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

解题步骤 1.8.2

化简 $R_{1}$ 。

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

解题步骤 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: $b_{11}, b_{22},$ and $b_{33}$

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

线性代数 示例

线性代数示例