대수 예제

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 4 & 9 & 12 \\ 1 & 7 & 11 & - 1 \end{array}]$

단계 1

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

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단계 1.1

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

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 4 & 9 & 12 \\ 1 - 1 & 7 - 3 & 11 - 2 & - 1 - 11 \end{array}]$

단계 1.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 4 & 9 & 12 \\ 0 & 4 & 9 & - 12 \end{array}]$

단계 2

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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단계 2.1

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ \frac{0}{4} & \frac{4}{4} & \frac{9}{4} & \frac{12}{4} \\ 0 & 4 & 9 & - 12 \end{array}]$

단계 2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 1 & \frac{9}{4} & 3 \\ 0 & 4 & 9 & - 12 \end{array}]$

단계 3

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

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단계 3.1

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

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 1 & \frac{9}{4} & 3 \\ 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & 9 - 4 (\frac{9}{4}) & - 12 - 4 \cdot 3 \end{array}]$

단계 3.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 1 & \frac{9}{4} & 3 \\ 0 & 0 & 0 & - 24 \end{array}]$

단계 4

Multiply each element of $R_{3}$ by $- \frac{1}{24}$ to make the entry at $3, 4$ a $1$ .

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단계 4.1

Multiply each element of $R_{3}$ by $- \frac{1}{24}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 1 & \frac{9}{4} & 3 \\ - \frac{1}{24} \cdot 0 & - \frac{1}{24} \cdot 0 & - \frac{1}{24} \cdot 0 & - \frac{1}{24} \cdot - 24 \end{array}]$

단계 4.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 1 & \frac{9}{4} & 3 \\ 0 & 0 & 0 & 1 \end{array}]$

단계 5

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

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단계 5.1

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

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 - 3 \cdot 0 & 1 - 3 \cdot 0 & \frac{9}{4} - 3 \cdot 0 & 3 - 3 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

단계 5.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & 1 & \frac{9}{4} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

단계 6

Perform the row operation $R_{1} = R_{1} - 11 R_{3}$ to make the entry at $1, 4$ a $0$ .

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단계 6.1

Perform the row operation $R_{1} = R_{1} - 11 R_{3}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 11 \cdot 0 & 3 - 11 \cdot 0 & 2 - 11 \cdot 0 & 11 - 11 \cdot 1 \\ 0 & 1 & \frac{9}{4} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

단계 6.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 3 & 2 & 0 \\ 0 & 1 & \frac{9}{4} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

단계 7

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

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단계 7.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & 2 - 3 (\frac{9}{4}) & 0 - 3 \cdot 0 \\ 0 & 1 & \frac{9}{4} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

단계 7.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & - \frac{19}{4} & 0 \\ 0 & 1 & \frac{9}{4} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$