선형 대수 예제

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

단계 1

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 1.1

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

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

단계 1.2

$R_{2}$ 을 간단히 합니다.

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

단계 2

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

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단계 2.1

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

$[\begin{array}{c} 1 & - 1 & 1 & - 1 \\ 0 & 3 & - 5 & 10 \\ 1 - 1 & - 2 + 1 & 3 - 1 & - 5 + 1 \end{array}]$

단계 2.2

$R_{3}$ 을 간단히 합니다.

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

단계 3

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

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단계 3.1

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

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

단계 3.2

$R_{2}$ 을 간단히 합니다.

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

단계 4

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

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단계 4.1

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

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

단계 4.2

$R_{3}$ 을 간단히 합니다.

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

단계 5

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

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단계 5.1

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

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

단계 5.2

$R_{3}$ 을 간단히 합니다.

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

단계 6

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

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단계 6.1

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

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

단계 6.2

$R_{2}$ 을 간단히 합니다.

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

단계 7

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

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단계 7.1

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

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

단계 7.2

$R_{1}$ 을 간단히 합니다.

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

단계 8

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

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단계 8.1

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

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

단계 8.2

$R_{1}$ 을 간단히 합니다.

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