线性代数示例

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

解题步骤 1

Find the determinant.

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解题步骤 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.

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解题步骤 1.1.1

Consider the corresponding sign chart.

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

解题步骤 1.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

解题步骤 1.1.3

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

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

解题步骤 1.1.4

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

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

解题步骤 1.1.5

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

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

解题步骤 1.1.6

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

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

解题步骤 1.1.7

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

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

解题步骤 1.1.8

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

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

解题步骤 1.1.9

Add the terms together.

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

解题步骤 1.2

将 $0$ 乘以 $| \begin{array}{c} 2 & 3 \\ 2 & 2 \end{array} |$ 。

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

解题步骤 1.3

计算 $| \begin{array}{c} 1 & - 1 \\ 2 & 2 \end{array} |$ 。

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解题步骤 1.3.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

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

解题步骤 1.3.2

化简行列式。

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解题步骤 1.3.2.1

化简每一项。

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解题步骤 1.3.2.1.1

将 $2$ 乘以 $1$ 。

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

解题步骤 1.3.2.1.2

将 $- 2$ 乘以 $- 1$ 。

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

解题步骤 1.3.2.2

将 $2$ 和 $2$ 相加。

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

解题步骤 1.4

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

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解题步骤 1.4.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

$1 \cdot 4 + 0 + 2 (2 \cdot - 1 - 1 \cdot 3)$

解题步骤 1.4.2

化简行列式。

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解题步骤 1.4.2.1

化简每一项。

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解题步骤 1.4.2.1.1

将 $2$ 乘以 $- 1$ 。

$1 \cdot 4 + 0 + 2 (- 2 - 1 \cdot 3)$

解题步骤 1.4.2.1.2

将 $- 1$ 乘以 $3$ 。

$1 \cdot 4 + 0 + 2 (- 2 - 3)$

解题步骤 1.4.2.2

从 $- 2$ 中减去 $3$ 。

$1 \cdot 4 + 0 + 2 \cdot - 5$

解题步骤 1.5

化简行列式。

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解题步骤 1.5.1

化简每一项。

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解题步骤 1.5.1.1

将 $4$ 乘以 $1$ 。

$4 + 0 + 2 \cdot - 5$

解题步骤 1.5.1.2

将 $2$ 乘以 $- 5$ 。

$4 + 0 - 10$

解题步骤 1.5.2

将 $4$ 和 $0$ 相加。

$4 - 10$

解题步骤 1.5.3

从 $4$ 中减去 $10$ 。

$- 6$

解题步骤 2

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

解题步骤 3

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

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

解题步骤 4

求行简化阶梯形矩阵。

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解题步骤 4.1

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

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解题步骤 4.1.1

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

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

解题步骤 4.1.2

化简 $R_{3}$ 。

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

解题步骤 4.2

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

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解题步骤 4.2.1

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

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

解题步骤 4.2.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & - 1 & 0 & 1 & 0 \\ 0 & 0 & - 6 & - 2 & 2 & 1 \end{array}]$

解题步骤 4.3

Multiply each element of $R_{3}$ by $- \frac{1}{6}$ to make the entry at $3, 3$ a $1$ .

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解题步骤 4.3.1

Multiply each element of $R_{3}$ by $- \frac{1}{6}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & - 1 & 0 & 1 & 0 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 2 & - \frac{1}{6} \cdot 2 & - \frac{1}{6} \cdot 1 \end{array}]$

解题步骤 4.3.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & - 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & \frac{1}{3} & - \frac{1}{3} & - \frac{1}{6} \end{array}]$

解题步骤 4.4

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

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解题步骤 4.4.1

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

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 0 + \frac{1}{3} & 1 - \frac{1}{3} & 0 - \frac{1}{6} \\ 0 & 0 & 1 & \frac{1}{3} & - \frac{1}{3} & - \frac{1}{6} \end{array}]$

解题步骤 4.4.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 2 & 3 & 1 & 0 & 0 \\ 0 & 1 & 0 & \frac{1}{3} & \frac{2}{3} & - \frac{1}{6} \\ 0 & 0 & 1 & \frac{1}{3} & - \frac{1}{3} & - \frac{1}{6} \end{array}]$

解题步骤 4.5

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

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解题步骤 4.5.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & 1 - 3 (\frac{1}{3}) & 0 - 3 (- \frac{1}{3}) & 0 - 3 (- \frac{1}{6}) \\ 0 & 1 & 0 & \frac{1}{3} & \frac{2}{3} & - \frac{1}{6} \\ 0 & 0 & 1 & \frac{1}{3} & - \frac{1}{3} & - \frac{1}{6} \end{array}]$

解题步骤 4.5.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 2 & 0 & 0 & 1 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{3} & \frac{2}{3} & - \frac{1}{6} \\ 0 & 0 & 1 & \frac{1}{3} & - \frac{1}{3} & - \frac{1}{6} \end{array}]$

解题步骤 4.6

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

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解题步骤 4.6.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 0 - 2 (\frac{1}{3}) & 1 - 2 (\frac{2}{3}) & \frac{1}{2} - 2 (- \frac{1}{6}) \\ 0 & 1 & 0 & \frac{1}{3} & \frac{2}{3} & - \frac{1}{6} \\ 0 & 0 & 1 & \frac{1}{3} & - \frac{1}{3} & - \frac{1}{6} \end{array}]$

解题步骤 4.6.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 0 & 0 & - \frac{2}{3} & - \frac{1}{3} & \frac{5}{6} \\ 0 & 1 & 0 & \frac{1}{3} & \frac{2}{3} & - \frac{1}{6} \\ 0 & 0 & 1 & \frac{1}{3} & - \frac{1}{3} & - \frac{1}{6} \end{array}]$

解题步骤 5

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

$[\begin{array}{c} - \frac{2}{3} & - \frac{1}{3} & \frac{5}{6} \\ \frac{1}{3} & \frac{2}{3} & - \frac{1}{6} \\ \frac{1}{3} & - \frac{1}{3} & - \frac{1}{6} \end{array}]$

线性代数 示例

线性代数示例