线性代数示例

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

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

解题步骤 1

将矩阵写成一个下三角矩阵和一个上三角矩阵的乘积。

[\begin{array}{c} 1 & 0 \\ l_{21} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} \\ 0 & u_{22} \end{array}] = [\begin{array}{c} 1 & - 1 \\ 2 & 3 \end{array}]

$[\begin{array}{c} 1 & 0 \\ l_{21} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} \\ 0 & u_{22} \end{array}] = [\begin{array}{c} 1 & - 1 \\ 2 & 3 \end{array}]$

解题步骤 2

乘以

[\begin{array}{c} 1 & 0 \\ l_{21} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} \\ 0 & u_{22} \end{array}]

$[\begin{array}{c} 1 & 0 \\ l_{21} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} \\ 0 & u_{22} \end{array}]$ 。

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解题步骤 2.1

当且仅当第一个矩阵的列数等于第二个矩阵的行数时，这两个矩阵才可以相乘。在本例中，第一个矩阵是

2 \times 2

$2 \times 2$ ，第二个矩阵是

2 \times 2

$2 \times 2$ 。

解题步骤 2.2

将第一个矩阵中的每一行乘以第二个矩阵中的每一列。

[\begin{array}{c} 1 u_{11} + 0 \cdot 0 & 1 u_{12} + 0 u_{22} \\ l_{21} u_{11} + 1 \cdot 0 & l_{21} u_{12} + 1 u_{22} \end{array}] = [\begin{array}{c} 1 & - 1 \\ 2 & 3 \end{array}]

$[\begin{array}{c} 1 u_{11} + 0 \cdot 0 & 1 u_{12} + 0 u_{22} \\ l_{21} u_{11} + 1 \cdot 0 & l_{21} u_{12} + 1 u_{22} \end{array}] = [\begin{array}{c} 1 & - 1 \\ 2 & 3 \end{array}]$

解题步骤 2.3

通过展开所有表达式化简矩阵的每一个元素。

[\begin{array}{c} u_{11} & u_{12} \\ l_{21} u_{11} & l_{21} u_{12} + u_{22} \end{array}] = [\begin{array}{c} 1 & - 1 \\ 2 & 3 \end{array}]

$[\begin{array}{c} u_{11} & u_{12} \\ l_{21} u_{11} & l_{21} u_{12} + u_{22} \end{array}] = [\begin{array}{c} 1 & - 1 \\ 2 & 3 \end{array}]$

[\begin{array}{c} u_{11} & u_{12} \\ l_{21} u_{11} & l_{21} u_{12} + u_{22} \end{array}] = [\begin{array}{c} 1 & - 1 \\ 2 & 3 \end{array}]

$[\begin{array}{c} u_{11} & u_{12} \\ l_{21} u_{11} & l_{21} u_{12} + u_{22} \end{array}] = [\begin{array}{c} 1 & - 1 \\ 2 & 3 \end{array}]$

解题步骤 3

求解。

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解题步骤 3.1

写成线性方程组。

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

l_{21} u_{11} = 2

$l_{21} u_{11} = 2$

l_{21} u_{12} + u_{22} = 3

$l_{21} u_{12} + u_{22} = 3$

解题步骤 3.2

求解方程组。

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解题步骤 3.2.1

将每个方程中所有出现的

u_{11}

$u_{11}$ 替换成

1

$1$ 。

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解题步骤 3.2.1.1

使用

1

$1$ 替换

l_{21} u_{11} = 2

$l_{21} u_{11} = 2$ 中所有出现的

u_{11}

$u_{11}$ .

l_{21} \cdot 1 = 2

$l_{21} \cdot 1 = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

l_{21} u_{12} + u_{22} = 3

$l_{21} u_{12} + u_{22} = 3$

解题步骤 3.2.1.2

化简左边。

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解题步骤 3.2.1.2.1

将

l_{21}

$l_{21}$ 乘以

1

$1$ 。

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

l_{21} u_{12} + u_{22} = 3

$l_{21} u_{12} + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

l_{21} u_{12} + u_{22} = 3

$l_{21} u_{12} + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

l_{21} u_{12} + u_{22} = 3

$l_{21} u_{12} + u_{22} = 3$

解题步骤 3.2.2

将每个方程中所有出现的

l_{21}

$l_{21}$ 替换成

2

$2$ 。

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解题步骤 3.2.2.1

使用

2

$2$ 替换

l_{21} u_{12} + u_{22} = 3

$l_{21} u_{12} + u_{22} = 3$ 中所有出现的

l_{21}

$l_{21}$ .

2 \cdot u_{12} + u_{22} = 3

$2 \cdot u_{12} + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

解题步骤 3.2.2.2

化简左边。

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解题步骤 3.2.2.2.1

将

2

$2$ 乘以

u_{12}

$u_{12}$ 。

2 u_{12} + u_{22} = 3

$2 u_{12} + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

2 u_{12} + u_{22} = 3

$2 u_{12} + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

2 u_{12} + u_{22} = 3

$2 u_{12} + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

解题步骤 3.2.3

将每个方程中所有出现的

u_{12}

$u_{12}$ 替换成

- 1

$- 1$ 。

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解题步骤 3.2.3.1

使用

- 1

$- 1$ 替换

2 u_{12} + u_{22} = 3

$2 u_{12} + u_{22} = 3$ 中所有出现的

u_{12}

$u_{12}$ .

2 (- 1) + u_{22} = 3

$2 (- 1) + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

解题步骤 3.2.3.2

化简左边。

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解题步骤 3.2.3.2.1

将

2

$2$ 乘以

- 1

$- 1$ 。

- 2 + u_{22} = 3

$- 2 + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

- 2 + u_{22} = 3

$- 2 + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

- 2 + u_{22} = 3

$- 2 + u_{22} = 3$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

解题步骤 3.2.4

将所有不包含

u_{22}

$u_{22}$ 的项移到等式右边。

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解题步骤 3.2.4.1

在等式两边都加上

2

$2$ 。

u_{22} = 3 + 2

$u_{22} = 3 + 2$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

解题步骤 3.2.4.2

将

3

$3$ 和

2

$2$ 相加。

u_{22} = 5

$u_{22} = 5$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

u_{22} = 5

$u_{22} = 5$

l_{21} = 2

$l_{21} = 2$

u_{11} = 1

$u_{11} = 1$

u_{12} = - 1

$u_{12} = - 1$

解题步骤 3.2.5

求解方程组。

u_{22} = 5 l_{21} = 2 u_{11} = 1 u_{12} = - 1

$u_{22} = 5 l_{21} = 2 u_{11} = 1 u_{12} = - 1$

解题步骤 3.2.6

列出所有解。

u_{22} = 5, l_{21} = 2, u_{11} = 1, u_{12} = - 1

$u_{22} = 5, l_{21} = 2, u_{11} = 1, u_{12} = - 1$

u_{22} = 5, l_{21} = 2, u_{11} = 1, u_{12} = - 1

$u_{22} = 5, l_{21} = 2, u_{11} = 1, u_{12} = - 1$

u_{22} = 5, l_{21} = 2, u_{11} = 1, u_{12} = - 1

$u_{22} = 5, l_{21} = 2, u_{11} = 1, u_{12} = - 1$

解题步骤 4

代入已求解的值。

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

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

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线性代数 示例

线性代数示例