유한 수학 예제

$[\begin{array}{c} 0.2 & 0 & - 0.02 \\ 0 & 0.5 & - 0.05 \\ 0 & - 0.5 & 0.15 \end{array}]$

단계 1

Find the determinant.

자세한 풀이 단계를 보려면 여기를 누르십시오...

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 column $1$ by its cofactor and add.

자세한 풀이 단계를 보려면 여기를 누르십시오...

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 $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 0.5 & - 0.05 \\ - 0.5 & 0.15 \end{array} |$

단계 1.1.4

Multiply element $a_{11}$ by its cofactor.

$0.2 | \begin{array}{c} 0.5 & - 0.05 \\ - 0.5 & 0.15 \end{array} |$

단계 1.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 0 & - 0.02 \\ - 0.5 & 0.15 \end{array} |$

단계 1.1.6

Multiply element $a_{21}$ by its cofactor.

$0 | \begin{array}{c} 0 & - 0.02 \\ - 0.5 & 0.15 \end{array} |$

단계 1.1.7

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

$| \begin{array}{c} 0 & - 0.02 \\ 0.5 & - 0.05 \end{array} |$

단계 1.1.8

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

$0 | \begin{array}{c} 0 & - 0.02 \\ 0.5 & - 0.05 \end{array} |$

단계 1.1.9

Add the terms together.

$0.2 | \begin{array}{c} 0.5 & - 0.05 \\ - 0.5 & 0.15 \end{array} | + 0 | \begin{array}{c} 0 & - 0.02 \\ - 0.5 & 0.15 \end{array} | + 0 | \begin{array}{c} 0 & - 0.02 \\ 0.5 & - 0.05 \end{array} |$

단계 1.2

$0$ 에 $| \begin{array}{c} 0 & - 0.02 \\ - 0.5 & 0.15 \end{array} |$ 을 곱합니다.

$0.2 | \begin{array}{c} 0.5 & - 0.05 \\ - 0.5 & 0.15 \end{array} | + 0 + 0 | \begin{array}{c} 0 & - 0.02 \\ 0.5 & - 0.05 \end{array} |$

단계 1.3

$0$ 에 $| \begin{array}{c} 0 & - 0.02 \\ 0.5 & - 0.05 \end{array} |$ 을 곱합니다.

$0.2 | \begin{array}{c} 0.5 & - 0.05 \\ - 0.5 & 0.15 \end{array} | + 0 + 0$

단계 1.4

$| \begin{array}{c} 0.5 & - 0.05 \\ - 0.5 & 0.15 \end{array} |$ 의 값을 구합니다.

자세한 풀이 단계를 보려면 여기를 누르십시오...

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.

$0.2 (0.5 \cdot 0.15 - (- 0.5 \cdot - 0.05)) + 0 + 0$

단계 1.4.2

행렬식을 간단히 합니다.

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 1.4.2.1

각 항을 간단히 합니다.

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 1.4.2.1.1

$0.5$ 에 $0.15$ 을 곱합니다.

$0.2 (0.075 - (- 0.5 \cdot - 0.05)) + 0 + 0$

단계 1.4.2.1.2

$- (- 0.5 \cdot - 0.05)$ 을 곱합니다.

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 1.4.2.1.2.1

$- 0.5$ 에 $- 0.05$ 을 곱합니다.

$0.2 (0.075 - 1 \cdot 0.025) + 0 + 0$

단계 1.4.2.1.2.2

$- 1$ 에 $0.025$ 을 곱합니다.

$0.2 (0.075 - 0.025) + 0 + 0$

단계 1.4.2.2

$0.075$ 에서 $0.025$ 을 뺍니다.

$0.2 \cdot 0.05 + 0 + 0$

단계 1.5

행렬식을 간단히 합니다.

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 1.5.1

$0.2$ 에 $0.05$ 을 곱합니다.

$0.01 + 0 + 0$

단계 1.5.2

$0.01$ 를 $0$ 에 더합니다.

$0.01 + 0$

단계 1.5.3

$0.01$ 를 $0$ 에 더합니다.

$0.01$

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.

$[\begin{array}{c} 0.2 & 0 & - 0.02 & 1 & 0 & 0 \\ 0 & 0.5 & - 0.05 & 0 & 1 & 0 \\ 0 & - 0.5 & 0.15 & 0 & 0 & 1 \end{array}]$

단계 4

기약 행 사다리꼴을 구합니다.

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 4.1

Multiply each element of $R_{1}$ by $\frac{1}{0.2}$ to make the entry at $1, 1$ a $1$ .

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 4.1.1

Multiply each element of $R_{1}$ by $\frac{1}{0.2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{0.2}{0.2} & \frac{0}{0.2} & \frac{- 0.02}{0.2} & \frac{1}{0.2} & \frac{0}{0.2} & \frac{0}{0.2} \\ 0 & 0.5 & - 0.05 & 0 & 1 & 0 \\ 0 & - 0.5 & 0.15 & 0 & 0 & 1 \end{array}]$

단계 4.1.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ 0 & 0.5 & - 0.05 & 0 & 1 & 0 \\ 0 & - 0.5 & 0.15 & 0 & 0 & 1 \end{array}]$

단계 4.2

Multiply each element of $R_{2}$ by $\frac{1}{0.5}$ to make the entry at $2, 2$ a $1$ .

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 4.2.1

Multiply each element of $R_{2}$ by $\frac{1}{0.5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ \frac{0}{0.5} & \frac{0.5}{0.5} & \frac{- 0.05}{0.5} & \frac{0}{0.5} & \frac{1}{0.5} & \frac{0}{0.5} \\ 0 & - 0.5 & 0.15 & 0 & 0 & 1 \end{array}]$

단계 4.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ 0 & 1 & - 0.1 & 0 & 2 & 0 \\ 0 & - 0.5 & 0.15 & 0 & 0 & 1 \end{array}]$

단계 4.3

Perform the row operation $R_{3} = R_{3} + 0.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 4.3.1

Perform the row operation $R_{3} = R_{3} + 0.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ 0 & 1 & - 0.1 & 0 & 2 & 0 \\ 0 + 0.5 \cdot 0 & - 0.5 + 0.5 \cdot 1 & 0.15 + 0.5 \cdot - 0.1 & 0 + 0.5 \cdot 0 & 0 + 0.5 \cdot 2 & 1 + 0.5 \cdot 0 \end{array}]$

단계 4.3.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ 0 & 1 & - 0.1 & 0 & 2 & 0 \\ 0 & 0 & 0.1 & 0 & 1 & 1 \end{array}]$

단계 4.4

Multiply each element of $R_{3}$ by $\frac{1}{0.1}$ to make the entry at $3, 3$ a $1$ .

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 4.4.1

Multiply each element of $R_{3}$ by $\frac{1}{0.1}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ 0 & 1 & - 0.1 & 0 & 2 & 0 \\ \frac{0}{0.1} & \frac{0}{0.1} & \frac{0.1}{0.1} & \frac{0}{0.1} & \frac{1}{0.1} & \frac{1}{0.1} \end{array}]$

단계 4.4.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ 0 & 1 & - 0.1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 10 & 10 \end{array}]$

단계 4.5

Perform the row operation $R_{2} = R_{2} + 0.1 R_{3}$ to make the entry at $2, 3$ a $0$ .

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 4.5.1

Perform the row operation $R_{2} = R_{2} + 0.1 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ 0 + 0.1 \cdot 0 & 1 + 0.1 \cdot 0 & - 0.1 + 0.1 \cdot 1 & 0 + 0.1 \cdot 0 & 2 + 0.1 \cdot 10 & 0 + 0.1 \cdot 10 \\ 0 & 0 & 1 & 0 & 10 & 10 \end{array}]$

단계 4.5.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & - 0.1 & 5 & 0 & 0 \\ 0 & 1 & 0 & 0 & 3 & 1 \\ 0 & 0 & 1 & 0 & 10 & 10 \end{array}]$

단계 4.6

Perform the row operation $R_{1} = R_{1} + 0.1 R_{3}$ to make the entry at $1, 3$ a $0$ .

자세한 풀이 단계를 보려면 여기를 누르십시오...

단계 4.6.1

Perform the row operation $R_{1} = R_{1} + 0.1 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 0.1 \cdot 0 & 0 + 0.1 \cdot 0 & - 0.1 + 0.1 \cdot 1 & 5 + 0.1 \cdot 0 & 0 + 0.1 \cdot 10 & 0 + 0.1 \cdot 10 \\ 0 & 1 & 0 & 0 & 3 & 1 \\ 0 & 0 & 1 & 0 & 10 & 10 \end{array}]$

단계 4.6.2

$R_{1}$ 을 간단히 합니다.

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

단계 5

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

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