线性代数示例

$[\begin{array}{c} 6 & - 4 & 10 \\ - 1 & 0 & 6 \\ 10 & 7 & - 4 \end{array}]$

解题步骤 1

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

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解题步骤 1.1

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

$[\begin{array}{c} \frac{6}{6} & \frac{- 4}{6} & \frac{10}{6} \\ - 1 & 0 & 6 \\ 10 & 7 & - 4 \end{array}]$

解题步骤 1.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ - 1 & 0 & 6 \\ 10 & 7 & - 4 \end{array}]$

解题步骤 2

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

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解题步骤 2.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ - 1 + 1 \cdot 1 & 0 - \frac{2}{3} & 6 + \frac{5}{3} \\ 10 & 7 & - 4 \end{array}]$

解题步骤 2.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & - \frac{2}{3} & \frac{23}{3} \\ 10 & 7 & - 4 \end{array}]$

解题步骤 3

Perform the row operation $R_{3} = R_{3} - 10 R_{1}$ to make the entry at $3, 1$ a $0$ .

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解题步骤 3.1

Perform the row operation $R_{3} = R_{3} - 10 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & - \frac{2}{3} & \frac{23}{3} \\ 10 - 10 \cdot 1 & 7 - 10 (- \frac{2}{3}) & - 4 - 10 (\frac{5}{3}) \end{array}]$

解题步骤 3.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & - \frac{2}{3} & \frac{23}{3} \\ 0 & \frac{41}{3} & - \frac{62}{3} \end{array}]$

解题步骤 4

Multiply each element of $R_{2}$ by $- \frac{3}{2}$ to make the entry at $2, 2$ a $1$ .

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解题步骤 4.1

Multiply each element of $R_{2}$ by $- \frac{3}{2}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ - \frac{3}{2} \cdot 0 & - \frac{3}{2} (- \frac{2}{3}) & - \frac{3}{2} \cdot \frac{23}{3} \\ 0 & \frac{41}{3} & - \frac{62}{3} \end{array}]$

解题步骤 4.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{23}{2} \\ 0 & \frac{41}{3} & - \frac{62}{3} \end{array}]$

解题步骤 5

Perform the row operation $R_{3} = R_{3} - \frac{41}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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解题步骤 5.1

Perform the row operation $R_{3} = R_{3} - \frac{41}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{23}{2} \\ 0 - \frac{41}{3} \cdot 0 & \frac{41}{3} - \frac{41}{3} \cdot 1 & - \frac{62}{3} - \frac{41}{3} (- \frac{23}{2}) \end{array}]$

解题步骤 5.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{23}{2} \\ 0 & 0 & \frac{273}{2} \end{array}]$

解题步骤 6

Multiply each element of $R_{3}$ by $\frac{2}{273}$ to make the entry at $3, 3$ a $1$ .

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解题步骤 6.1

Multiply each element of $R_{3}$ by $\frac{2}{273}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{23}{2} \\ \frac{2}{273} \cdot 0 & \frac{2}{273} \cdot 0 & \frac{2}{273} \cdot \frac{273}{2} \end{array}]$

解题步骤 6.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & 1 & - \frac{23}{2} \\ 0 & 0 & 1 \end{array}]$

解题步骤 7

Perform the row operation $R_{2} = R_{2} + \frac{23}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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解题步骤 7.1

Perform the row operation $R_{2} = R_{2} + \frac{23}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 + \frac{23}{2} \cdot 0 & 1 + \frac{23}{2} \cdot 0 & - \frac{23}{2} + \frac{23}{2} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

解题步骤 7.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{5}{3} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 8

Perform the row operation $R_{1} = R_{1} - \frac{5}{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

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解题步骤 8.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{3} \cdot 0 & - \frac{2}{3} - \frac{5}{3} \cdot 0 & \frac{5}{3} - \frac{5}{3} \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 8.2

化简 $R_{1}$ 。

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

解题步骤 9

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

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解题步骤 9.1

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

$[\begin{array}{c} 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & 0 + \frac{2}{3} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 9.2

化简 $R_{1}$ 。

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

线性代数 示例

线性代数示例