代数示例

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

解题步骤 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

$3$ by its cofactor and add.

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解题步骤 1.1

Consider the corresponding sign chart.

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

解题步骤 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 1.3

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

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

解题步骤 1.4

Multiply element

$a_{13}$ by its cofactor.

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

解题步骤 1.5

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

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

解题步骤 1.6

Multiply element

$a_{23}$ by its cofactor.

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

解题步骤 1.7

The minor for

$a_{33}$ is the determinant with row

$3$ and column

$3$ deleted.

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

解题步骤 1.8

Multiply element

$a_{33}$ by its cofactor.

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

解题步骤 1.9

Add the terms together.

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

解题步骤 2

将

$0$ 乘以

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

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

解题步骤 3

将

$0$ 乘以

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

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

解题步骤 4

计算

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

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解题步骤 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。

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

解题步骤 4.2

化简行列式。

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解题步骤 4.2.1

化简每一项。

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解题步骤 4.2.1.1

将

$1$ 乘以

$1$ 。

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

解题步骤 4.2.1.2

将

$- 1$ 乘以

$1$ 。

$2 (1 - 1) + 0 + 0$

解题步骤 4.2.2

从

$1$ 中减去

$1$ 。

$2 \cdot 0 + 0 + 0$

解题步骤 5

化简行列式。

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解题步骤 5.1

将

$2$ 乘以

$0$ 。

$0 + 0 + 0$

解题步骤 5.2

将

$0$ 和

$0$ 相加。

$0 + 0$

解题步骤 5.3

将

$0$ 和

$0$ 相加。

$0$

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