有限数学示例

$[\begin{array}{c} 0.01 & 0.34 & 0.25 & 0.23 \\ 0.16 & 0.13 & 0.03 & 0.02 \\ 0.12 & 0.05 & 0.06 & 0.01 \\ 0.14 & 0.08 & 0.15 & 0.28 \end{array}]$

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 $\frac{1}{0.01}$ 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 $\frac{1}{0.01}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{0.01}{0.01} & \frac{0.34}{0.01} & \frac{0.25}{0.01} & \frac{0.23}{0.01} \\ 0.16 & 0.13 & 0.03 & 0.02 \\ 0.12 & 0.05 & 0.06 & 0.01 \\ 0.14 & 0.08 & 0.15 & 0.28 \end{array}]$

解题步骤 1.1.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0.16 & 0.13 & 0.03 & 0.02 \\ 0.12 & 0.05 & 0.06 & 0.01 \\ 0.14 & 0.08 & 0.15 & 0.28 \end{array}]$

解题步骤 1.2

Perform the row operation $R_{2} = R_{2} - 0.16 R_{1}$ to make the entry at $2, 1$ a $0$ .

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解题步骤 1.2.1

Perform the row operation $R_{2} = R_{2} - 0.16 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0.16 - 0.16 \cdot 1 & 0.13 - 0.16 \cdot 34 & 0.03 - 0.16 \cdot 25 & 0.02 - 0.16 \cdot 23 \\ 0.12 & 0.05 & 0.06 & 0.01 \\ 0.14 & 0.08 & 0.15 & 0.28 \end{array}]$

解题步骤 1.2.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & - 5.31 & - 3.97 & - 3.66 \\ 0.12 & 0.05 & 0.06 & 0.01 \\ 0.14 & 0.08 & 0.15 & 0.28 \end{array}]$

解题步骤 1.3

Perform the row operation $R_{3} = R_{3} - 0.12 R_{1}$ to make the entry at $3, 1$ a $0$ .

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解题步骤 1.3.1

Perform the row operation $R_{3} = R_{3} - 0.12 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & - 5.31 & - 3.97 & - 3.66 \\ 0.12 - 0.12 \cdot 1 & 0.05 - 0.12 \cdot 34 & 0.06 - 0.12 \cdot 25 & 0.01 - 0.12 \cdot 23 \\ 0.14 & 0.08 & 0.15 & 0.28 \end{array}]$

解题步骤 1.3.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & - 5.31 & - 3.97 & - 3.66 \\ 0 & - 4.03 & - 2.94 & - 2.75 \\ 0.14 & 0.08 & 0.15 & 0.28 \end{array}]$

解题步骤 1.4

Perform the row operation $R_{4} = R_{4} - 0.14 R_{1}$ to make the entry at $4, 1$ a $0$ .

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解题步骤 1.4.1

Perform the row operation $R_{4} = R_{4} - 0.14 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & - 5.31 & - 3.97 & - 3.66 \\ 0 & - 4.03 & - 2.94 & - 2.75 \\ 0.14 - 0.14 \cdot 1 & 0.08 - 0.14 \cdot 34 & 0.15 - 0.14 \cdot 25 & 0.28 - 0.14 \cdot 23 \end{array}]$

解题步骤 1.4.2

化简 $R_{4}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & - 5.31 & - 3.97 & - 3.66 \\ 0 & - 4.03 & - 2.94 & - 2.75 \\ 0 & - 4.68 & - 3.35 & - 2.94 \end{array}]$

解题步骤 1.5

Multiply each element of $R_{2}$ by $\frac{1}{- 5.31}$ to make the entry at $2, 2$ a $1$ .

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解题步骤 1.5.1

Multiply each element of $R_{2}$ by $\frac{1}{- 5.31}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ \frac{0}{- 5.31} & \frac{- 5.31}{- 5.31} & \frac{- 3.97}{- 5.31} & \frac{- 3.66}{- 5.31} \\ 0 & - 4.03 & - 2.94 & - 2.75 \\ 0 & - 4.68 & - 3.35 & - 2.94 \end{array}]$

解题步骤 1.5.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & - 4.03 & - 2.94 & - 2.75 \\ 0 & - 4.68 & - 3.35 & - 2.94 \end{array}]$

解题步骤 1.6

Perform the row operation $R_{3} = R_{3} + 4.03 R_{2}$ to make the entry at $3, 2$ a $0$ .

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解题步骤 1.6.1

Perform the row operation $R_{3} = R_{3} + 4.03 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 + 4.03 \cdot 0 & - 4.03 + 4.03 \cdot 1 & - 2.94 + 4.03 \cdot 0.74764595 & - 2.75 + 4.03 \cdot 0.68926553 \\ 0 & - 4.68 & - 3.35 & - 2.94 \end{array}]$

解题步骤 1.6.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 0.07301318 & 0.02774011 \\ 0 & - 4.68 & - 3.35 & - 2.94 \end{array}]$

解题步骤 1.7

Perform the row operation $R_{4} = R_{4} + 4.68 R_{2}$ to make the entry at $4, 2$ a $0$ .

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解题步骤 1.7.1

Perform the row operation $R_{4} = R_{4} + 4.68 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 0.07301318 & 0.02774011 \\ 0 + 4.68 \cdot 0 & - 4.68 + 4.68 \cdot 1 & - 3.35 + 4.68 \cdot 0.74764595 & - 2.94 + 4.68 \cdot 0.68926553 \end{array}]$

解题步骤 1.7.2

化简 $R_{4}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 0.07301318 & 0.02774011 \\ 0 & 0 & 0.14898305 & 0.28576271 \end{array}]$

解题步骤 1.8

Multiply each element of $R_{3}$ by $\frac{1}{0.07301318}$ to make the entry at $3, 3$ a $1$ .

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解题步骤 1.8.1

Multiply each element of $R_{3}$ by $\frac{1}{0.07301318}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ \frac{0}{0.07301318} & \frac{0}{0.07301318} & \frac{0.07301318}{0.07301318} & \frac{0.02774011}{0.07301318} \\ 0 & 0 & 0.14898305 & 0.28576271 \end{array}]$

解题步骤 1.8.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 1 & 0.37993293 \\ 0 & 0 & 0.14898305 & 0.28576271 \end{array}]$

解题步骤 1.9

Perform the row operation $R_{4} = R_{4} - 0.14898305 R_{3}$ to make the entry at $4, 3$ a $0$ .

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解题步骤 1.9.1

Perform the row operation $R_{4} = R_{4} - 0.14898305 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 1 & 0.37993293 \\ 0 - 0.14898305 \cdot 0 & 0 - 0.14898305 \cdot 0 & 0.14898305 - 0.14898305 \cdot 1 & 0.28576271 - 0.14898305 \cdot 0.37993293 \end{array}]$

解题步骤 1.9.2

化简 $R_{4}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 1 & 0.37993293 \\ 0 & 0 & 0 & 0.22915914 \end{array}]$

解题步骤 1.10

Multiply each element of $R_{4}$ by $\frac{1}{0.22915914}$ to make the entry at $4, 4$ a $1$ .

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解题步骤 1.10.1

Multiply each element of $R_{4}$ by $\frac{1}{0.22915914}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 1 & 0.37993293 \\ \frac{0}{0.22915914} & \frac{0}{0.22915914} & \frac{0}{0.22915914} & \frac{0.22915914}{0.22915914} \end{array}]$

解题步骤 1.10.2

化简 $R_{4}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 1 & 0.37993293 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.11

Perform the row operation $R_{3} = R_{3} - 0.37993293 R_{4}$ to make the entry at $3, 4$ a $0$ .

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解题步骤 1.11.1

Perform the row operation $R_{3} = R_{3} - 0.37993293 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 - 0.37993293 \cdot 0 & 0 - 0.37993293 \cdot 0 & 1 - 0.37993293 \cdot 0 & 0.37993293 - 0.37993293 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.11.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0.68926553 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.12

Perform the row operation $R_{2} = R_{2} - 0.68926553 R_{4}$ to make the entry at $2, 4$ a $0$ .

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解题步骤 1.12.1

Perform the row operation $R_{2} = R_{2} - 0.68926553 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 - 0.68926553 \cdot 0 & 1 - 0.68926553 \cdot 0 & 0.74764595 - 0.68926553 \cdot 0 & 0.68926553 - 0.68926553 \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.12.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 23 \\ 0 & 1 & 0.74764595 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.13

Perform the row operation $R_{1} = R_{1} - 23 R_{4}$ to make the entry at $1, 4$ a $0$ .

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解题步骤 1.13.1

Perform the row operation $R_{1} = R_{1} - 23 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 23 \cdot 0 & 34 - 23 \cdot 0 & 25 - 23 \cdot 0 & 23 - 23 \cdot 1 \\ 0 & 1 & 0.74764595 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.13.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 0 \\ 0 & 1 & 0.74764595 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.14

Perform the row operation $R_{2} = R_{2} - 0.74764595 R_{3}$ to make the entry at $2, 3$ a $0$ .

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解题步骤 1.14.1

Perform the row operation $R_{2} = R_{2} - 0.74764595 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 34 & 25 & 0 \\ 0 - 0.74764595 \cdot 0 & 1 - 0.74764595 \cdot 0 & 0.74764595 - 0.74764595 \cdot 1 & 0 - 0.74764595 \cdot 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.14.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 34 & 25 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.15

Perform the row operation $R_{1} = R_{1} - 25 R_{3}$ to make the entry at $1, 3$ a $0$ .

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解题步骤 1.15.1

Perform the row operation $R_{1} = R_{1} - 25 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 25 \cdot 0 & 34 - 25 \cdot 0 & 25 - 25 \cdot 1 & 0 - 25 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.15.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 34 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.16

Perform the row operation $R_{1} = R_{1} - 34 R_{2}$ to make the entry at $1, 2$ a $0$ .

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解题步骤 1.16.1

Perform the row operation $R_{1} = R_{1} - 34 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 34 \cdot 0 & 34 - 34 \cdot 1 & 0 - 34 \cdot 0 & 0 - 34 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 1.16.2

化简 $R_{1}$ 。

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

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: $a_{11}, a_{22}, a_{33},$ and $a_{44}$

Pivot Columns: $1, 2, 3,$ and $4$

有限数学 示例

有限数学示例