线性代数示例

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

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 $1$ 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.

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

解题步骤 1.4

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

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

解题步骤 1.5

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

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

解题步骤 1.6

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

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

解题步骤 1.7

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

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

解题步骤 1.8

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

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

解题步骤 1.9

Add the terms together.

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

解题步骤 2

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

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

解题步骤 3

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

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

解题步骤 4

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

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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。

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

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 $0$ 乘以 $0$ 。

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

解题步骤 4.2.1.2

将 $- 1$ 乘以 $1$ 。

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

解题步骤 4.2.2

从 $0$ 中减去 $1$ 。

$1 \cdot - 1 + 0 + 0$

解题步骤 5

化简行列式。

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解题步骤 5.1

将 $- 1$ 乘以 $1$ 。

$- 1 + 0 + 0$

解题步骤 5.2

将 $- 1$ 和 $0$ 相加。

$- 1 + 0$

解题步骤 5.3

将 $- 1$ 和 $0$ 相加。

$- 1$

线性代数 示例

线性代数示例