대수 예제

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

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 $3$ 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 $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 1 & 4 \\ 2 & 1 \end{array} |$

단계 1.1.4

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} 1 & 4 \\ 2 & 1 \end{array} |$

단계 1.1.5

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} 3 & 4 \\ 1 & 1 \end{array} |$

단계 1.1.6

Multiply element $a_{32}$ by its cofactor.

$- 1 | \begin{array}{c} 3 & 4 \\ 1 & 1 \end{array} |$

단계 1.1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

$| \begin{array}{c} 3 & 1 \\ 1 & 2 \end{array} |$

단계 1.1.8

Multiply element $a_{33}$ by its cofactor.

$0 | \begin{array}{c} 3 & 1 \\ 1 & 2 \end{array} |$

단계 1.1.9

Add the terms together.

$0 | \begin{array}{c} 1 & 4 \\ 2 & 1 \end{array} | - 1 | \begin{array}{c} 3 & 4 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 1 \\ 1 & 2 \end{array} |$

단계 1.2

$0$ 에 $| \begin{array}{c} 1 & 4 \\ 2 & 1 \end{array} |$ 을 곱합니다.

$0 - 1 | \begin{array}{c} 3 & 4 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 1 \\ 1 & 2 \end{array} |$

단계 1.3

$0$ 에 $| \begin{array}{c} 3 & 1 \\ 1 & 2 \end{array} |$ 을 곱합니다.

$0 - 1 | \begin{array}{c} 3 & 4 \\ 1 & 1 \end{array} | + 0$

단계 1.4

$| \begin{array}{c} 3 & 4 \\ 1 & 1 \end{array} |$ 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.

$0 - 1 (3 \cdot 1 - 1 \cdot 4) + 0$

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

$3$ 에 $1$ 을 곱합니다.

$0 - 1 (3 - 1 \cdot 4) + 0$

단계 1.4.2.1.2

$- 1$ 에 $4$ 을 곱합니다.

$0 - 1 (3 - 4) + 0$

단계 1.4.2.2

$3$ 에서 $4$ 을 뺍니다.

$0 - 1 \cdot - 1 + 0$

단계 1.5

행렬식을 간단히 합니다.

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단계 1.5.1

$- 1$ 에 $- 1$ 을 곱합니다.

$0 + 1 + 0$

단계 1.5.2

$0$ 를 $1$ 에 더합니다.

$1 + 0$

단계 1.5.3

$1$ 를 $0$ 에 더합니다.

$1$

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.

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

단계 4

기약 행 사다리꼴을 구합니다.

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단계 4.1

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

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단계 4.1.1

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

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

단계 4.1.2

$R_{1}$ 을 간단히 합니다.

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

단계 4.2

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

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단계 4.2.1

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

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

단계 4.2.2

$R_{2}$ 을 간단히 합니다.

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

단계 4.3

Multiply each element of $R_{2}$ by $\frac{3}{5}$ to make the entry at $2, 2$ a $1$ .

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단계 4.3.1

Multiply each element of $R_{2}$ by $\frac{3}{5}$ to make the entry at $2, 2$ a $1$ .

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

단계 4.3.2

$R_{2}$ 을 간단히 합니다.

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

단계 4.4

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

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단계 4.4.1

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

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

단계 4.4.2

$R_{3}$ 을 간단히 합니다.

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

단계 4.5

Multiply each element of $R_{3}$ by $5$ to make the entry at $3, 3$ a $1$ .

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단계 4.5.1

Multiply each element of $R_{3}$ by $5$ to make the entry at $3, 3$ a $1$ .

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

단계 4.5.2

$R_{3}$ 을 간단히 합니다.

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

단계 4.6

Perform the row operation $R_{2} = R_{2} + \frac{1}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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단계 4.6.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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

단계 4.6.2

$R_{2}$ 을 간단히 합니다.

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

단계 4.7

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

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단계 4.7.1

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

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

단계 4.7.2

$R_{1}$ 을 간단히 합니다.

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

단계 4.8

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 4.8.1

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

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

단계 4.8.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & 0 & - 1 & 4 & - 7 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & - 3 & 5 \end{array}]$

단계 5

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

$[\begin{array}{c} - 1 & 4 & - 7 \\ 0 & 0 & 1 \\ 1 & - 3 & 5 \end{array}]$

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