선형 대수 예제

$[\begin{array}{c} 1 & - 4 & 2 \\ - 3 & 13 & 1 \\ - 6 & 8 & h \end{array}]$

단계 1

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단계 1.1

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

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단계 1.1.1

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

$[\begin{array}{c} 1 & - 4 & 2 \\ - 3 + 3 \cdot 1 & 13 + 3 \cdot - 4 & 1 + 3 \cdot 2 \\ - 6 & 8 & h \end{array}]$

단계 1.1.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & 7 \\ - 6 & 8 & h \end{array}]$

단계 1.2

Perform the row operation $R_{3} = R_{3} + 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

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단계 1.2.1

Perform the row operation $R_{3} = R_{3} + 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & 7 \\ - 6 + 6 \cdot 1 & 8 + 6 \cdot - 4 & h + 6 \cdot 2 \end{array}]$

단계 1.2.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & 7 \\ 0 & - 16 & h + 12 \end{array}]$

단계 1.3

Perform the row operation $R_{3} = R_{3} + 16 R_{2}$ to make the entry at $3, 2$ a $0$ .

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단계 1.3.1

Perform the row operation $R_{3} = R_{3} + 16 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & 7 \\ 0 + 16 \cdot 0 & - 16 + 16 \cdot 1 & h + 12 + 16 \cdot 7 \end{array}]$

단계 1.3.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & 7 \\ 0 & 0 & h + 124 \end{array}]$

단계 1.4

Multiply each element of $R_{3}$ by $\frac{1}{h + 124}$ to make the entry at $3, 3$ a $1$ .

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단계 1.4.1

Multiply each element of $R_{3}$ by $\frac{1}{h + 124}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & 7 \\ \frac{0}{h + 124} & \frac{0}{h + 124} & \frac{h + 124}{h + 124} \end{array}]$

단계 1.4.2

$R_{3}$ 을 간단히 합니다.

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

단계 1.5

Perform the row operation $R_{2} = R_{2} - 7 R_{3}$ to make the entry at $2, 3$ a $0$ .

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단계 1.5.1

Perform the row operation $R_{2} = R_{2} - 7 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 - 7 \cdot 0 & 1 - 7 \cdot 0 & 7 - 7 \cdot 1 \\ 0 & 0 & 1 \end{array}]$

단계 1.5.2

$R_{2}$ 을 간단히 합니다.

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

단계 1.6

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

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단계 1.6.1

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

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

단계 1.6.2

$R_{1}$ 을 간단히 합니다.

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

단계 1.7

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

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단계 1.7.1

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

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

단계 1.7.2

$R_{1}$ 을 간단히 합니다.

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

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: $a_{11}, a_{22},$ and $a_{33}$

Pivot Columns: $1, 2,$ and $3$