有限数学示例

$r^{2} + (- 3) r = [\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 3 & 5 & 8 & 1 \\ - 2 & 7 & 8 & 9 \end{array}]$

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

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解题步骤 1.1.1

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

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 3 - 3 \cdot 1 & 5 - 3 \cdot 9 & 8 - 3 \cdot - 4 & 1 - 3 \cdot - 7 \\ - 2 & 7 & 8 & 9 \end{array}]$

解题步骤 1.1.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & - 22 & 20 & 22 \\ - 2 & 7 & 8 & 9 \end{array}]$

解题步骤 1.2

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

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解题步骤 1.2.1

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

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & - 22 & 20 & 22 \\ - 2 + 2 \cdot 1 & 7 + 2 \cdot 9 & 8 + 2 \cdot - 4 & 9 + 2 \cdot - 7 \end{array}]$

解题步骤 1.2.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & - 22 & 20 & 22 \\ 0 & 25 & 0 & - 5 \end{array}]$

解题步骤 1.3

Multiply each element of $R_{2}$ by $- \frac{1}{22}$ 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 $- \frac{1}{22}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ - \frac{1}{22} \cdot 0 & - \frac{1}{22} \cdot - 22 & - \frac{1}{22} \cdot 20 & - \frac{1}{22} \cdot 22 \\ 0 & 25 & 0 & - 5 \end{array}]$

解题步骤 1.3.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & 1 & - \frac{10}{11} & - 1 \\ 0 & 25 & 0 & - 5 \end{array}]$

解题步骤 1.4

Perform the row operation $R_{3} = R_{3} - 25 R_{2}$ to make the entry at $3, 2$ a $0$ .

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解题步骤 1.4.1

Perform the row operation $R_{3} = R_{3} - 25 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & 1 & - \frac{10}{11} & - 1 \\ 0 - 25 \cdot 0 & 25 - 25 \cdot 1 & 0 - 25 (- \frac{10}{11}) & - 5 - 25 \cdot - 1 \end{array}]$

解题步骤 1.4.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & 1 & - \frac{10}{11} & - 1 \\ 0 & 0 & \frac{250}{11} & 20 \end{array}]$

解题步骤 1.5

Multiply each element of $R_{3}$ by $\frac{11}{250}$ 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 $\frac{11}{250}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & 1 & - \frac{10}{11} & - 1 \\ \frac{11}{250} \cdot 0 & \frac{11}{250} \cdot 0 & \frac{11}{250} \cdot \frac{250}{11} & \frac{11}{250} \cdot 20 \end{array}]$

解题步骤 1.5.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & 1 & - \frac{10}{11} & - 1 \\ 0 & 0 & 1 & \frac{22}{25} \end{array}]$

解题步骤 1.6

Perform the row operation $R_{2} = R_{2} + \frac{10}{11} R_{3}$ to make the entry at $2, 3$ a $0$ .

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解题步骤 1.6.1

Perform the row operation $R_{2} = R_{2} + \frac{10}{11} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 + \frac{10}{11} \cdot 0 & 1 + \frac{10}{11} \cdot 0 & - \frac{10}{11} + \frac{10}{11} \cdot 1 & - 1 + \frac{10}{11} \cdot \frac{22}{25} \\ 0 & 0 & 1 & \frac{22}{25} \end{array}]$

解题步骤 1.6.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 9 & - 4 & - 7 \\ 0 & 1 & 0 & - \frac{1}{5} \\ 0 & 0 & 1 & \frac{22}{25} \end{array}]$

解题步骤 1.7

Perform the row operation $R_{1} = R_{1} + 4 R_{3}$ to make the entry at $1, 3$ a $0$ .

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解题步骤 1.7.1

Perform the row operation $R_{1} = R_{1} + 4 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 4 \cdot 0 & 9 + 4 \cdot 0 & - 4 + 4 \cdot 1 & - 7 + 4 (\frac{22}{25}) \\ 0 & 1 & 0 & - \frac{1}{5} \\ 0 & 0 & 1 & \frac{22}{25} \end{array}]$

解题步骤 1.7.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 9 & 0 & - \frac{87}{25} \\ 0 & 1 & 0 & - \frac{1}{5} \\ 0 & 0 & 1 & \frac{22}{25} \end{array}]$

解题步骤 1.8

Perform the row operation $R_{1} = R_{1} - 9 R_{2}$ to make the entry at $1, 2$ a $0$ .

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解题步骤 1.8.1

Perform the row operation $R_{1} = R_{1} - 9 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 9 \cdot 0 & 9 - 9 \cdot 1 & 0 - 9 \cdot 0 & - \frac{87}{25} - 9 (- \frac{1}{5}) \\ 0 & 1 & 0 & - \frac{1}{5} \\ 0 & 0 & 1 & \frac{22}{25} \end{array}]$

解题步骤 1.8.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 0 & 0 & - \frac{42}{25} \\ 0 & 1 & 0 & - \frac{1}{5} \\ 0 & 0 & 1 & \frac{22}{25} \end{array}]$

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