有限数学示例

$[\begin{array}{c} 0.75 & 0.15 & 0.1 \\ 0.25 & 0.65 & 0.1 \\ 0.1 & 0.2 & 0.7 \end{array}]$

解题步骤 1

求行简化阶梯形矩阵。

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解题步骤 1.1

Multiply each element of $R_{1}$ by $\frac{1}{0.75}$ to make the entry at $1, 1$ a $1$ .

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解题步骤 1.1.1

Multiply each element of $R_{1}$ by $\frac{1}{0.75}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{0.75}{0.75} & \frac{0.15}{0.75} & \frac{0.1}{0.75} \\ 0.25 & 0.65 & 0.1 \\ 0.1 & 0.2 & 0.7 \end{array}]$

解题步骤 1.1.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0.25 & 0.65 & 0.1 \\ 0.1 & 0.2 & 0.7 \end{array}]$

解题步骤 1.2

Perform the row operation $R_{2} = R_{2} - 0.25 R_{1}$ to make the entry at $2, 1$ a $0$ .

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解题步骤 1.2.1

Perform the row operation $R_{2} = R_{2} - 0.25 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0.25 - 0.25 \cdot 1 & 0.65 - 0.25 \cdot 0.2 & 0.1 - 0.25 \cdot 0.1\overline{3} \\ 0.1 & 0.2 & 0.7 \end{array}]$

解题步骤 1.2.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 0.6 & 0.0\overline{6} \\ 0.1 & 0.2 & 0.7 \end{array}]$

解题步骤 1.3

Perform the row operation $R_{3} = R_{3} - 0.1 R_{1}$ to make the entry at $3, 1$ a $0$ .

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解题步骤 1.3.1

Perform the row operation $R_{3} = R_{3} - 0.1 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 0.6 & 0.0\overline{6} \\ 0.1 - 0.1 \cdot 1 & 0.2 - 0.1 \cdot 0.2 & 0.7 - 0.1 \cdot 0.1\overline{3} \end{array}]$

解题步骤 1.3.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 0.6 & 0.0\overline{6} \\ 0 & 0.18 & 0.68\overline{6} \end{array}]$

解题步骤 1.4

Multiply each element of $R_{2}$ by $\frac{1}{0.6}$ to make the entry at $2, 2$ a $1$ .

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解题步骤 1.4.1

Multiply each element of $R_{2}$ by $\frac{1}{0.6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ \frac{0}{0.6} & \frac{0.6}{0.6} & \frac{0.0\overline{6}}{0.6} \\ 0 & 0.18 & 0.68\overline{6} \end{array}]$

解题步骤 1.4.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 1 & 0.\overline{1} \\ 0 & 0.18 & 0.68\overline{6} \end{array}]$

解题步骤 1.5

Perform the row operation $R_{3} = R_{3} - 0.18 R_{2}$ to make the entry at $3, 2$ a $0$ .

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解题步骤 1.5.1

Perform the row operation $R_{3} = R_{3} - 0.18 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 1 & 0.\overline{1} \\ 0 - 0.18 \cdot 0 & 0.18 - 0.18 \cdot 1 & 0.68\overline{6} - 0.18 \cdot 0.\overline{1} \end{array}]$

解题步骤 1.5.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 1 & 0.\overline{1} \\ 0 & 0 & 0.\overline{6} \end{array}]$

解题步骤 1.6

Multiply each element of $R_{3}$ by $\frac{1}{0.\overline{6}}$ to make the entry at $3, 3$ a $1$ .

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解题步骤 1.6.1

Multiply each element of $R_{3}$ by $\frac{1}{0.\overline{6}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 1 & 0.\overline{1} \\ \frac{0}{0.\overline{6}} & \frac{0}{0.\overline{6}} & \frac{0.\overline{6}}{0.\overline{6}} \end{array}]$

解题步骤 1.6.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 1 & 0.\overline{1} \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.7

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

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解题步骤 1.7.1

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

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 - 0.\overline{1} \cdot 0 & 1 - 0.\overline{1} \cdot 0 & 0.\overline{1} - 0.\overline{1} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.7.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 0.2 & 0.1\overline{3} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.8

Perform the row operation $R_{1} = R_{1} - 0.1\overline{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

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解题步骤 1.8.1

Perform the row operation $R_{1} = R_{1} - 0.1\overline{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0.1\overline{3} \cdot 0 & 0.2 - 0.1\overline{3} \cdot 0 & 0.1\overline{3} - 0.1\overline{3} \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.8.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 0.2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.9

Perform the row operation $R_{1} = R_{1} - 0.2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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解题步骤 1.9.1

Perform the row operation $R_{1} = R_{1} - 0.2 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0.2 \cdot 0 & 0.2 - 0.2 \cdot 1 & 0 - 0.2 \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 1.9.2

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

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

有限数学 示例

有限数学示例