初级微积分示例

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

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

$1$ 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

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

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

解题步骤 1.4

Multiply element

$a_{11}$ by its cofactor.

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

解题步骤 1.5

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

解题步骤 1.6

Multiply element

$a_{21}$ by its cofactor.

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

解题步骤 1.7

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

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

解题步骤 1.8

Multiply element

$a_{31}$ by its cofactor.

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

解题步骤 1.9

The minor for

$a_{41}$ is the determinant with row

$4$ and column

$1$ deleted.

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

解题步骤 1.10

Multiply element

$a_{41}$ by its cofactor.

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

解题步骤 1.11

Add the terms together.

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

解题步骤 2

将

$0$ 乘以

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

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

解题步骤 3

将

$0$ 乘以

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

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

解题步骤 4

计算

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

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解题步骤 4.1

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

$2$ by its cofactor and add.

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解题步骤 4.1.1

Consider the corresponding sign chart.

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

解题步骤 4.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

解题步骤 4.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

解题步骤 4.1.4

Multiply element

$a_{21}$ by its cofactor.

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

解题步骤 4.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

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

解题步骤 4.1.6

Multiply element

$a_{22}$ by its cofactor.

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

解题步骤 4.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

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

解题步骤 4.1.8

Multiply element

$a_{23}$ by its cofactor.

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

解题步骤 4.1.9

Add the terms together.

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

解题步骤 4.2

将

$0$ 乘以

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

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

解题步骤 4.3

将

$0$ 乘以

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

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

解题步骤 4.4

计算

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

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解题步骤 4.4.1

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。

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

解题步骤 4.4.2

化简行列式。

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解题步骤 4.4.2.1

化简每一项。

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解题步骤 4.4.2.1.1

将

$1$ 乘以

$1$ 。

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

解题步骤 4.4.2.1.2

将

$- 2$ 乘以

$1$ 。

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

解题步骤 4.4.2.2

从

$1$ 中减去

$2$ 。

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

解题步骤 4.5

化简行列式。

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解题步骤 4.5.1

将

$- 1$ 乘以

$1$ 。

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

解题步骤 4.5.2

从

$0$ 中减去

$1$ 。

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

解题步骤 4.5.3

将

$- 1$ 和

$0$ 相加。

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

解题步骤 5

计算

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

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解题步骤 5.1

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

$2$ by its cofactor and add.

点击获取更多步骤...

解题步骤 5.1.1

Consider the corresponding sign chart.

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

解题步骤 5.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

解题步骤 5.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

解题步骤 5.1.4

Multiply element

$a_{21}$ by its cofactor.

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

解题步骤 5.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

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

解题步骤 5.1.6

Multiply element

$a_{22}$ by its cofactor.

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

解题步骤 5.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

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

解题步骤 5.1.8

Multiply element

$a_{23}$ by its cofactor.

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

解题步骤 5.1.9

Add the terms together.

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

解题步骤 5.2

将

$0$ 乘以

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

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

解题步骤 5.3

计算

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

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解题步骤 5.3.1

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。

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

解题步骤 5.3.2

化简行列式。

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解题步骤 5.3.2.1

化简每一项。

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解题步骤 5.3.2.1.1

将

$2$ 乘以

$1$ 。

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

解题步骤 5.3.2.1.2

将

$- 1$ 乘以

$1$ 。

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

解题步骤 5.3.2.2

从

$2$ 中减去

$1$ 。

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

解题步骤 5.4

计算

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

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解题步骤 5.4.1

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。

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

解题步骤 5.4.2

化简行列式。

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解题步骤 5.4.2.1

化简每一项。

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解题步骤 5.4.2.1.1

将

$1$ 乘以

$1$ 。

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

解题步骤 5.4.2.1.2

将

$- 2$ 乘以

$2$ 。

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

解题步骤 5.4.2.2

从

$1$ 中减去

$4$ 。

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

解题步骤 5.5

化简行列式。

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解题步骤 5.5.1

化简每一项。

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解题步骤 5.5.1.1

将

$- 1$ 乘以

$1$ 。

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

解题步骤 5.5.1.2

将

$- 2$ 乘以

$- 3$ 。

$0 - 1 \cdot - 1 + 1 (- 1 + 0 + 6) + 0$

解题步骤 5.5.2

将

$- 1$ 和

$0$ 相加。

$0 - 1 \cdot - 1 + 1 (- 1 + 6) + 0$

解题步骤 5.5.3

将

$- 1$ 和

$6$ 相加。

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

解题步骤 6

化简行列式。

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解题步骤 6.1

化简每一项。

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解题步骤 6.1.1

将

$- 1$ 乘以

$- 1$ 。

$0 + 1 + 1 \cdot 5 + 0$

解题步骤 6.1.2

将

$5$ 乘以

$1$ 。

$0 + 1 + 5 + 0$

解题步骤 6.2

将

$0$ 和

$1$ 相加。

$1 + 5 + 0$

解题步骤 6.3

将

$1$ 和

$5$ 相加。

$6 + 0$

解题步骤 6.4

将

$6$ 和

$0$ 相加。

$6$

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初级微积分 示例

初级微积分示例