선형 대수 예제

$[\begin{array}{c} 3 & 1 & 0 \\ 2 & 1 & - 4 \\ 2 & - 1 & 20 \end{array}]$

단계 1

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단계 1.1

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

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단계 1.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{1}{3} & \frac{0}{3} \\ 2 & 1 & - 4 \\ 2 & - 1 & 20 \end{array}]$

단계 1.1.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 2 & 1 & - 4 \\ 2 & - 1 & 20 \end{array}]$

단계 1.2

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

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단계 1.2.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 2 - 2 \cdot 1 & 1 - 2 (\frac{1}{3}) & - 4 - 2 \cdot 0 \\ 2 & - 1 & 20 \end{array}]$

단계 1.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 0 & \frac{1}{3} & - 4 \\ 2 & - 1 & 20 \end{array}]$

단계 1.3

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

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단계 1.3.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 0 & \frac{1}{3} & - 4 \\ 2 - 2 \cdot 1 & - 1 - 2 (\frac{1}{3}) & 20 - 2 \cdot 0 \end{array}]$

단계 1.3.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 0 & \frac{1}{3} & - 4 \\ 0 & - \frac{5}{3} & 20 \end{array}]$

단계 1.4

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

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단계 1.4.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 3 \cdot 0 & 3 (\frac{1}{3}) & 3 \cdot - 4 \\ 0 & - \frac{5}{3} & 20 \end{array}]$

단계 1.4.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 0 & 1 & - 12 \\ 0 & - \frac{5}{3} & 20 \end{array}]$

단계 1.5

Perform the row operation $R_{3} = R_{3} + \frac{5}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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단계 1.5.1

Perform the row operation $R_{3} = R_{3} + \frac{5}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 0 & 1 & - 12 \\ 0 + \frac{5}{3} \cdot 0 & - \frac{5}{3} + \frac{5}{3} \cdot 1 & 20 + \frac{5}{3} \cdot - 12 \end{array}]$

단계 1.5.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 0 & 1 & - 12 \\ 0 & 0 & 0 \end{array}]$

단계 1.6

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

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단계 1.6.1

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

$[\begin{array}{c} 1 - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & 0 - \frac{1}{3} \cdot - 12 \\ 0 & 1 & - 12 \\ 0 & 0 & 0 \end{array}]$

단계 1.6.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & 4 \\ 0 & 1 & - 12 \\ 0 & 0 & 0 \end{array}]$

단계 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 $2$

단계 3

The rank is the number of pivot columns.

$2$