线性代数示例

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

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 row $1$ 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.

$| \begin{array}{c} f & 1 \\ 1 & f \end{array} |$

解题步骤 1.1.4

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

$f | \begin{array}{c} f & 1 \\ 1 & f \end{array} |$

解题步骤 1.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

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

解题步骤 1.1.6

Multiply element $a_{12}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1 \\ 0 & f \end{array} |$

解题步骤 1.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

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

解题步骤 1.1.8

Multiply element $a_{13}$ by its cofactor.

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

解题步骤 1.1.9

Add the terms together.

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

解题步骤 1.2

将 $0$ 乘以 $| \begin{array}{c} 1 & f \\ 0 & 1 \end{array} |$ 。

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

解题步骤 1.3

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

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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。

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

解题步骤 1.3.2

化简每一项。

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解题步骤 1.3.2.1

将 $f$ 乘以 $f$ 。

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

解题步骤 1.3.2.2

将 $- 1$ 乘以 $1$ 。

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

解题步骤 1.4

计算 $| \begin{array}{c} 1 & 1 \\ 0 & f \end{array} |$ 。

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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。

$f (f^{2} - 1) - 1 (1 f + 0 \cdot 1) + 0$

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 $f$ 乘以 $1$ 。

$f (f^{2} - 1) - 1 (f + 0 \cdot 1) + 0$

解题步骤 1.4.2.1.2

将 $0$ 乘以 $1$ 。

$f (f^{2} - 1) - 1 (f + 0) + 0$

解题步骤 1.4.2.2

将 $f$ 和 $0$ 相加。

$f (f^{2} - 1) - 1 f + 0$

解题步骤 1.5

化简行列式。

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解题步骤 1.5.1

将 $f (f^{2} - 1) - 1 f$ 和 $0$ 相加。

$f (f^{2} - 1) - 1 f$

解题步骤 1.5.2

化简每一项。

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解题步骤 1.5.2.1

运用分配律。

$f \cdot f^{2} + f \cdot - 1 - 1 f$

解题步骤 1.5.2.2

通过指数相加将 $f$ 乘以 $f^{2}$ 。

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解题步骤 1.5.2.2.1

将 $f$ 乘以 $f^{2}$ 。

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解题步骤 1.5.2.2.1.1

对 $f$ 进行 $1$ 次方运算。

$f^{1} f^{2} + f \cdot - 1 - 1 f$

解题步骤 1.5.2.2.1.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$f^{1 + 2} + f \cdot - 1 - 1 f$

解题步骤 1.5.2.2.2

将 $1$ 和 $2$ 相加。

$f^{3} + f \cdot - 1 - 1 f$

解题步骤 1.5.2.3

将 $- 1$ 移到 $f$ 的左侧。

$f^{3} - 1 \cdot f - 1 f$

解题步骤 1.5.2.4

将 $- 1 f$ 重写为 $- f$ 。

$f^{3} - f - 1 f$

解题步骤 1.5.2.5

将 $- 1 f$ 重写为 $- f$ 。

$f^{3} - f - f$

解题步骤 1.5.3

从 $- f$ 中减去 $f$ 。

$f^{3} - 2 f$

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.

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

解题步骤 4

求行简化阶梯形矩阵。

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解题步骤 4.1

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

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解题步骤 4.1.1

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

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

解题步骤 4.1.2

化简 $R_{1}$ 。

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

解题步骤 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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解题步骤 4.2.1

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

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

解题步骤 4.2.2

化简 $R_{2}$ 。

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

解题步骤 4.3

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

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解题步骤 4.3.1

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

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

解题步骤 4.3.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & \frac{1}{f} & 0 & \frac{1}{f} & 0 & 0 \\ 0 & 1 & \frac{f}{(f + 1) (f - 1)} & - \frac{1}{(f + 1) (f - 1)} & \frac{f}{(f + 1) (f - 1)} & 0 \\ 0 & 1 & f & 0 & 0 & 1 \end{array}]$

解题步骤 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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解题步骤 4.4.1

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

$[\begin{array}{c} 1 & \frac{1}{f} & 0 & \frac{1}{f} & 0 & 0 \\ 0 & 1 & \frac{f}{(f + 1) (f - 1)} & - \frac{1}{(f + 1) (f - 1)} & \frac{f}{(f + 1) (f - 1)} & 0 \\ 0 - 0 & 1 - 1 & f - \frac{f}{(f + 1) (f - 1)} & 0 + \frac{1}{(f + 1) (f - 1)} & 0 - \frac{f}{(f + 1) (f - 1)} & 1 - 0 \end{array}]$

解题步骤 4.4.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & \frac{1}{f} & 0 & \frac{1}{f} & 0 & 0 \\ 0 & 1 & \frac{f}{(f + 1) (f - 1)} & - \frac{1}{(f + 1) (f - 1)} & \frac{f}{(f + 1) (f - 1)} & 0 \\ 0 & 0 & f - \frac{f}{(f + 1) (f - 1)} & \frac{1}{(f + 1) (f - 1)} & - \frac{f}{(f + 1) (f - 1)} & 1 \end{array}]$

解题步骤 4.5

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

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解题步骤 4.5.1

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

$[\begin{array}{c} 1 & \frac{1}{f} & 0 & \frac{1}{f} & 0 & 0 \\ 0 & 1 & \frac{f}{(f + 1) (f - 1)} & - \frac{1}{(f + 1) (f - 1)} & \frac{f}{(f + 1) (f - 1)} & 0 \\ \frac{0}{f - \frac{f}{(f + 1) (f - 1)}} & \frac{0}{f - \frac{f}{(f + 1) (f - 1)}} & \frac{f - \frac{f}{(f + 1) (f - 1)}}{f - \frac{f}{(f + 1) (f - 1)}} & \frac{\frac{1}{(f + 1) (f - 1)}}{f - \frac{f}{(f + 1) (f - 1)}} & \frac{- \frac{f}{(f + 1) (f - 1)}}{f - \frac{f}{(f + 1) (f - 1)}} & \frac{1}{f - \frac{f}{(f + 1) (f - 1)}} \end{array}]$

解题步骤 4.5.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & \frac{1}{f} & 0 & \frac{1}{f} & 0 & 0 \\ 0 & 1 & \frac{f}{(f + 1) (f - 1)} & - \frac{1}{(f + 1) (f - 1)} & \frac{f}{(f + 1) (f - 1)} & 0 \\ 0 & 0 & 1 & \frac{1}{f (f^{2} - 2)} & - \frac{1}{f^{2} - 2} & \frac{(f + 1) (f - 1)}{f (f^{2} - 2)} \end{array}]$

解题步骤 4.6

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

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解题步骤 4.6.1

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

$[\begin{array}{c} 1 & \frac{1}{f} & 0 & \frac{1}{f} & 0 & 0 \\ 0 - \frac{f}{(f + 1) (f - 1)} \cdot 0 & 1 - \frac{f}{(f + 1) (f - 1)} \cdot 0 & \frac{f}{(f + 1) (f - 1)} - \frac{f}{(f + 1) (f - 1)} \cdot 1 & - \frac{1}{(f + 1) (f - 1)} - \frac{f}{(f + 1) (f - 1)} \cdot \frac{1}{f (f^{2} - 2)} & \frac{f}{(f + 1) (f - 1)} - \frac{f}{(f + 1) (f - 1)} (- \frac{1}{f^{2} - 2}) & 0 - \frac{f}{(f + 1) (f - 1)} \cdot \frac{(f + 1) (f - 1)}{f (f^{2} - 2)} \\ 0 & 0 & 1 & \frac{1}{f (f^{2} - 2)} & - \frac{1}{f^{2} - 2} & \frac{(f + 1) (f - 1)}{f (f^{2} - 2)} \end{array}]$

解题步骤 4.6.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & \frac{1}{f} & 0 & \frac{1}{f} & 0 & 0 \\ 0 & 1 & 0 & - \frac{1}{(f + 1) (f - 1)} - \frac{1}{(f + 1) (f - 1) (f^{2} - 2)} & \frac{f}{(f + 1) (f - 1)} + \frac{f}{(f + 1) (f - 1) (f^{2} - 2)} & - \frac{1}{f^{2} - 2} \\ 0 & 0 & 1 & \frac{1}{f (f^{2} - 2)} & - \frac{1}{f^{2} - 2} & \frac{(f + 1) (f - 1)}{f (f^{2} - 2)} \end{array}]$

解题步骤 4.7

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

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解题步骤 4.7.1

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

$[\begin{array}{c} 1 - \frac{1}{f} \cdot 0 & \frac{1}{f} - \frac{1}{f} \cdot 1 & 0 - \frac{1}{f} \cdot 0 & \frac{1}{f} - \frac{1}{f} (- \frac{1}{(f + 1) (f - 1)} - \frac{1}{(f + 1) (f - 1) (f^{2} - 2)}) & 0 - \frac{1}{f} (\frac{f}{(f + 1) (f - 1)} + \frac{f}{(f + 1) (f - 1) (f^{2} - 2)}) & 0 - \frac{1}{f} (- \frac{1}{f^{2} - 2}) \\ 0 & 1 & 0 & - \frac{1}{(f + 1) (f - 1)} - \frac{1}{(f + 1) (f - 1) (f^{2} - 2)} & \frac{f}{(f + 1) (f - 1)} + \frac{f}{(f + 1) (f - 1) (f^{2} - 2)} & - \frac{1}{f^{2} - 2} \\ 0 & 0 & 1 & \frac{1}{f (f^{2} - 2)} & - \frac{1}{f^{2} - 2} & \frac{(f + 1) (f - 1)}{f (f^{2} - 2)} \end{array}]$

解题步骤 4.7.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 0 & 0 & \frac{1}{f} + \frac{1}{f (f + 1) (f - 1)} + \frac{1}{f (f + 1) (f - 1) (f^{2} - 2)} & - \frac{1}{(f + 1) (f - 1)} - \frac{1}{(f + 1) (f - 1) (f^{2} - 2)} & \frac{1}{f (f^{2} - 2)} \\ 0 & 1 & 0 & - \frac{1}{(f + 1) (f - 1)} - \frac{1}{(f + 1) (f - 1) (f^{2} - 2)} & \frac{f}{(f + 1) (f - 1)} + \frac{f}{(f + 1) (f - 1) (f^{2} - 2)} & - \frac{1}{f^{2} - 2} \\ 0 & 0 & 1 & \frac{1}{f (f^{2} - 2)} & - \frac{1}{f^{2} - 2} & \frac{(f + 1) (f - 1)}{f (f^{2} - 2)} \end{array}]$

解题步骤 5

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

$[\begin{array}{c} \frac{1}{f} + \frac{1}{f (f + 1) (f - 1)} + \frac{1}{f (f + 1) (f - 1) (f^{2} - 2)} & - \frac{1}{(f + 1) (f - 1)} - \frac{1}{(f + 1) (f - 1) (f^{2} - 2)} & \frac{1}{f (f^{2} - 2)} \\ - \frac{1}{(f + 1) (f - 1)} - \frac{1}{(f + 1) (f - 1) (f^{2} - 2)} & \frac{f}{(f + 1) (f - 1)} + \frac{f}{(f + 1) (f - 1) (f^{2} - 2)} & - \frac{1}{f^{2} - 2} \\ \frac{1}{f (f^{2} - 2)} & - \frac{1}{f^{2} - 2} & \frac{(f + 1) (f - 1)}{f (f^{2} - 2)} \end{array}]$

线性代数 示例

线性代数示例