선형 대수 예제

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

단계 1

Find the determinant.

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단계 1.1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in row

$2$ by its cofactor and add.

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단계 1.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

단계 1.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

단계 1.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} 0 & 3 \\ 2 & 4 \end{array} |$

단계 1.1.4

Multiply element

$a_{21}$ by its cofactor.

$- 3 | \begin{array}{c} 0 & 3 \\ 2 & 4 \end{array} |$

단계 1.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

$| \begin{array}{c} 2 & 3 \\ 0 & 4 \end{array} |$

단계 1.1.6

Multiply element

$a_{22}$ by its cofactor.

$0 | \begin{array}{c} 2 & 3 \\ 0 & 4 \end{array} |$

단계 1.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

$| \begin{array}{c} 2 & 0 \\ 0 & 2 \end{array} |$

단계 1.1.8

Multiply element

$a_{23}$ by its cofactor.

$0 | \begin{array}{c} 2 & 0 \\ 0 & 2 \end{array} |$

단계 1.1.9

Add the terms together.

$- 3 | \begin{array}{c} 0 & 3 \\ 2 & 4 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 0 & 4 \end{array} | + 0 | \begin{array}{c} 2 & 0 \\ 0 & 2 \end{array} |$

단계 1.2

$0$ 에

$| \begin{array}{c} 2 & 3 \\ 0 & 4 \end{array} |$ 을 곱합니다.

$- 3 | \begin{array}{c} 0 & 3 \\ 2 & 4 \end{array} | + 0 + 0 | \begin{array}{c} 2 & 0 \\ 0 & 2 \end{array} |$

단계 1.3

$0$ 에

$| \begin{array}{c} 2 & 0 \\ 0 & 2 \end{array} |$ 을 곱합니다.

$- 3 | \begin{array}{c} 0 & 3 \\ 2 & 4 \end{array} | + 0 + 0$

단계 1.4

$| \begin{array}{c} 0 & 3 \\ 2 & 4 \end{array} |$ 의 값을 구합니다.

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단계 1.4.1

$2 \times 2$ 행렬의 행렬식은

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 공식을 이용해 계산합니다.

$- 3 (0 \cdot 4 - 2 \cdot 3) + 0 + 0$

단계 1.4.2

행렬식을 간단히 합니다.

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단계 1.4.2.1

각 항을 간단히 합니다.

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단계 1.4.2.1.1

$0$ 에

$4$ 을 곱합니다.

$- 3 (0 - 2 \cdot 3) + 0 + 0$

단계 1.4.2.1.2

$- 2$ 에

$3$ 을 곱합니다.

$- 3 (0 - 6) + 0 + 0$

단계 1.4.2.2

$0$ 에서

$6$ 을 뺍니다.

$- 3 \cdot - 6 + 0 + 0$

단계 1.5

행렬식을 간단히 합니다.

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단계 1.5.1

$- 3$ 에

$- 6$ 을 곱합니다.

$18 + 0 + 0$

단계 1.5.2

$18$ 를

$0$ 에 더합니다.

$18 + 0$

단계 1.5.3

$18$ 를

$0$ 에 더합니다.

$18$

단계 2

Since the determinant is non-zero, the inverse exists.

단계 3

Set up a

$3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

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

단계 4

기약 행 사다리꼴을 구합니다.

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단계 4.1

Multiply each element of

$R_{1}$ by

$\frac{1}{2}$ to make the entry at

$1, 1$ a

$1$ .

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단계 4.1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{2}$ to make the entry at

$1, 1$ a

$1$ .

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

단계 4.1.2

$R_{1}$ 을 간단히 합니다.

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

단계 4.2

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 4.2.1

Perform the row operation

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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

단계 4.2.2

$R_{2}$ 을 간단히 합니다.

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

단계 4.3

Swap

$R_{3}$ with

$R_{2}$ to put a nonzero entry at

$2, 2$ .

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

단계 4.4

Multiply each element of

$R_{2}$ by

$\frac{1}{2}$ to make the entry at

$2, 2$ a

$1$ .

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단계 4.4.1

Multiply each element of

$R_{2}$ by

$\frac{1}{2}$ to make the entry at

$2, 2$ a

$1$ .

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

단계 4.4.2

$R_{2}$ 을 간단히 합니다.

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

단계 4.5

Multiply each element of

$R_{3}$ by

$- \frac{2}{9}$ to make the entry at

$3, 3$ a

$1$ .

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단계 4.5.1

Multiply each element of

$R_{3}$ by

$- \frac{2}{9}$ to make the entry at

$3, 3$ a

$1$ .

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

단계 4.5.2

$R_{3}$ 을 간단히 합니다.

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

단계 4.6

Perform the row operation

$R_{2} = R_{2} - 2 R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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단계 4.6.1

Perform the row operation

$R_{2} = R_{2} - 2 R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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

단계 4.6.2

$R_{2}$ 을 간단히 합니다.

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

단계 4.7

Perform the row operation

$R_{1} = R_{1} - \frac{3}{2} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

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단계 4.7.1

Perform the row operation

$R_{1} = R_{1} - \frac{3}{2} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

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

단계 4.7.2

$R_{1}$ 을 간단히 합니다.

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

단계 5

The right half of the reduced row echelon form is the inverse.

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

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