线性代数示例

$[\begin{array}{c} - 4 & 1 \\ 0 & 1 \\ 4 & 1 \\ 8 & 1 \end{array}]$

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 $- \frac{1}{4}$ 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 $- \frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{4} \cdot - 4 & - \frac{1}{4} \cdot 1 \\ 0 & 1 \\ 4 & 1 \\ 8 & 1 \end{array}]$

解题步骤 1.1.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & - \frac{1}{4} \\ 0 & 1 \\ 4 & 1 \\ 8 & 1 \end{array}]$

解题步骤 1.2

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

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解题步骤 1.2.1

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

$[\begin{array}{c} 1 & - \frac{1}{4} \\ 0 & 1 \\ 4 - 4 \cdot 1 & 1 - 4 (- \frac{1}{4}) \\ 8 & 1 \end{array}]$

解题步骤 1.2.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & - \frac{1}{4} \\ 0 & 1 \\ 0 & 2 \\ 8 & 1 \end{array}]$

解题步骤 1.3

Perform the row operation $R_{4} = R_{4} - 8 R_{1}$ to make the entry at $4, 1$ a $0$ .

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解题步骤 1.3.1

Perform the row operation $R_{4} = R_{4} - 8 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} \\ 0 & 1 \\ 0 & 2 \\ 8 - 8 \cdot 1 & 1 - 8 (- \frac{1}{4}) \end{array}]$

解题步骤 1.3.2

化简 $R_{4}$ 。

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

解题步骤 1.4

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

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解题步骤 1.4.1

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

$[\begin{array}{c} 1 & - \frac{1}{4} \\ 0 & 1 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 \\ 0 & 3 \end{array}]$

解题步骤 1.4.2

化简 $R_{3}$ 。

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

解题步骤 1.5

Perform the row operation $R_{4} = R_{4} - 3 R_{2}$ to make the entry at $4, 2$ a $0$ .

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解题步骤 1.5.1

Perform the row operation $R_{4} = R_{4} - 3 R_{2}$ to make the entry at $4, 2$ a $0$ .

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

解题步骤 1.5.2

化简 $R_{4}$ 。

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

解题步骤 1.6

Perform the row operation $R_{1} = R_{1} + \frac{1}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

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解题步骤 1.6.1

Perform the row operation $R_{1} = R_{1} + \frac{1}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{1}{4} \cdot 0 & - \frac{1}{4} + \frac{1}{4} \cdot 1 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{array}]$

解题步骤 1.6.2

化简 $R_{1}$ 。

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

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

Pivot Columns: $1$ and $2$

线性代数 示例

线性代数示例