代数示例

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

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。

$[\begin{array}{c} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

解题步骤 2

乘以 $[\begin{array}{c} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}]$ 。

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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 $3 \times 3$ ，第二个矩阵是 $3 \times 3$ 。

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。

$[\begin{array}{c} 1 u_{11} + 0 \cdot 0 + 0 \cdot 0 & 1 u_{12} + 0 u_{22} + 0 \cdot 0 & 1 u_{13} + 0 u_{23} + 0 u_{33} \\ l_{21} u_{11} + 1 \cdot 0 + 0 \cdot 0 & l_{21} u_{12} + 1 u_{22} + 0 \cdot 0 & l_{21} u_{13} + 1 u_{23} + 0 u_{33} \\ l_{31} u_{11} + l_{32} \cdot 0 + 1 \cdot 0 & l_{31} u_{12} + l_{32} u_{22} + 1 \cdot 0 & l_{31} u_{13} + l_{32} u_{23} + 1 u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

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。

$[\begin{array}{c} u_{11} & u_{12} & u_{13} \\ l_{21} u_{11} & l_{21} u_{12} + u_{22} & l_{21} u_{13} + u_{23} \\ l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

解题步骤 3

求解。

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解题步骤 3.1

写成线性方程组。

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{11} = 3$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{11} = 1$

$l_{31} u_{12} + l_{32} u_{22} = - 2$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

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 $u_{11}$ 替换成 $1$ 。

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解题步骤 3.2.1.1

使用 $1$ 替换 $l_{21} u_{11} = 3$ 中所有出现的 $u_{11}$ .

$l_{21} \cdot 1 = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{11} = 1$

$l_{31} u_{12} + l_{32} u_{22} = - 2$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

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 $l_{21}$ 乘以 $1$ 。

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{11} = 1$

$l_{31} u_{12} + l_{32} u_{22} = - 2$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{11} = 1$

$l_{31} u_{12} + l_{32} u_{22} = - 2$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

解题步骤 3.2.1.3

使用 $1$ 替换 $l_{31} u_{11} = 1$ 中所有出现的 $u_{11}$ .

$l_{31} \cdot 1 = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{12} + l_{32} u_{22} = - 2$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

解题步骤 3.2.1.4

化简左边。

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解题步骤 3.2.1.4.1

将 $l_{31}$ 乘以 $1$ 。

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{12} + l_{32} u_{22} = - 2$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{12} + l_{32} u_{22} = - 2$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{12} + l_{32} u_{22} = - 2$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

解题步骤 3.2.2

将每个方程中所有出现的 $l_{31}$ 替换成 $1$ 。

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解题步骤 3.2.2.1

使用 $1$ 替换 $l_{31} u_{12} + l_{32} u_{22} = - 2$ 中所有出现的 $l_{31}$ .

$1 \cdot u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

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 $u_{12}$ 乘以 $1$ 。

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

解题步骤 3.2.2.3

使用 $1$ 替换 $l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$ 中所有出现的 $l_{31}$ .

$1 \cdot u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

解题步骤 3.2.2.4

化简左边。

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解题步骤 3.2.2.4.1

将 $u_{13}$ 乘以 $1$ 。

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

解题步骤 3.2.3

将每个方程中所有出现的 $l_{21}$ 替换成 $3$ 。

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解题步骤 3.2.3.1

使用 $3$ 替换 $l_{21} u_{12} + u_{22} = 1$ 中所有出现的 $l_{21}$ .

$3 \cdot u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{13} + u_{23} = - 3$

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 $3$ 乘以 $u_{12}$ 。

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{13} + u_{23} = - 3$

解题步骤 3.2.3.3

使用 $3$ 替换 $l_{21} u_{13} + u_{23} = - 3$ 中所有出现的 $l_{21}$ .

$3 \cdot u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.3.4

化简左边。

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解题步骤 3.2.3.4.1

将 $3$ 乘以 $u_{13}$ 。

$3 u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$3 u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$3 u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.4

将每个方程中所有出现的 $u_{12}$ 替换成 $1$ 。

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解题步骤 3.2.4.1

使用 $1$ 替换 $3 u_{12} + u_{22} = 1$ 中所有出现的 $u_{12}$ .

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

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.4.2

化简左边。

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解题步骤 3.2.4.2.1

将 $3$ 乘以 $1$ 。

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.4.3

使用 $1$ 替换 $u_{12} + l_{32} u_{22} = - 2$ 中所有出现的 $u_{12}$ .

$1 + l_{32} u_{22} = - 2$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.4.4

化简左边。

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解题步骤 3.2.4.4.1

去掉圆括号。

$1 + l_{32} u_{22} = - 2$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 + l_{32} u_{22} = - 2$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 + l_{32} u_{22} = - 2$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.5

将每个方程中所有出现的 $u_{13}$ 替换成 $1$ 。

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解题步骤 3.2.5.1

将所有不包含 $u_{22}$ 的项移到等式右边。

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解题步骤 3.2.5.1.1

从等式两边同时减去 $3$ 。

$u_{22} = 1 - 3$

$1 + l_{32} u_{22} = - 2$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.5.1.2

从 $1$ 中减去 $3$ 。

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.5.2

使用 $1$ 替换 $3 u_{13} + u_{23} = - 3$ 中所有出现的 $u_{13}$ .

$3 (1) + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.5.3

化简左边。

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解题步骤 3.2.5.3.1

将 $3$ 乘以 $1$ 。

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.5.4

使用 $1$ 替换 $u_{13} + l_{32} u_{23} + u_{33} = - 5$ 中所有出现的 $u_{13}$ .

$1 + l_{32} u_{23} + u_{33} = - 5$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.5.5

化简左边。

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解题步骤 3.2.5.5.1

去掉圆括号。

$1 + l_{32} u_{23} + u_{33} = - 5$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 + l_{32} u_{23} + u_{33} = - 5$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 + l_{32} u_{23} + u_{33} = - 5$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.6

将每个方程中所有出现的 $u_{22}$ 替换成 $- 2$ 。

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解题步骤 3.2.6.1

将所有不包含 $u_{23}$ 的项移到等式右边。

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解题步骤 3.2.6.1.1

从等式两边同时减去 $3$ 。

$u_{23} = - 3 - 3$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.6.1.2

从 $- 3$ 中减去 $3$ 。

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.6.2

使用 $- 2$ 替换 $1 + l_{32} u_{22} = - 2$ 中所有出现的 $u_{22}$ .

$1 + l_{32} \cdot - 2 = - 2$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.6.3

化简左边。

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解题步骤 3.2.6.3.1

将 $- 2$ 移到 $l_{32}$ 的左侧。

$1 - 2 l_{32} = - 2$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 2 l_{32} = - 2$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 2 l_{32} = - 2$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.7

将每个方程中所有出现的 $u_{23}$ 替换成 $- 6$ 。

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解题步骤 3.2.7.1

将所有不包含 $l_{32}$ 的项移到等式右边。

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解题步骤 3.2.7.1.1

从等式两边同时减去 $1$ 。

$- 2 l_{32} = - 2 - 1$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.7.1.2

从 $- 2$ 中减去 $1$ 。

$- 2 l_{32} = - 3$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 2 l_{32} = - 3$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.7.2

将 $- 2 l_{32} = - 3$ 中的每一项除以 $- 2$ 并化简。

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解题步骤 3.2.7.2.1

将 $- 2 l_{32} = - 3$ 中的每一项都除以 $- 2$ 。

$\frac{- 2 l_{32}}{- 2} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.7.2.2

化简左边。

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解题步骤 3.2.7.2.2.1

约去 $- 2$ 的公因数。

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解题步骤 3.2.7.2.2.1.1

约去公因数。

$\frac{- 2 l_{32}}{- 2} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.7.2.2.1.2

用 $l_{32}$ 除以 $1$ 。

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.7.2.3

化简右边。

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解题步骤 3.2.7.2.3.1

将两个负数相除得到一个正数。

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.7.3

使用 $- 6$ 替换 $1 + l_{32} u_{23} + u_{33} = - 5$ 中所有出现的 $u_{23}$ .

$1 + l_{32} \cdot - 6 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.7.4

化简左边。

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解题步骤 3.2.7.4.1

将 $- 6$ 移到 $l_{32}$ 的左侧。

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.8

将每个方程中所有出现的 $l_{32}$ 替换成 $\frac{3}{2}$ 。

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解题步骤 3.2.8.1

使用 $\frac{3}{2}$ 替换 $1 - 6 l_{32} + u_{33} = - 5$ 中所有出现的 $l_{32}$ .

$1 - 6 (\frac{3}{2}) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.8.2

化简左边。

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解题步骤 3.2.8.2.1

化简 $1 - 6 (\frac{3}{2}) + u_{33}$ 。

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解题步骤 3.2.8.2.1.1

化简每一项。

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解题步骤 3.2.8.2.1.1.1

约去 $2$ 的公因数。

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解题步骤 3.2.8.2.1.1.1.1

从 $- 6$ 中分解出因数 $2$ 。

$1 + 2 (- 3) (\frac{3}{2}) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.8.2.1.1.1.2

约去公因数。

$1 + 2 \cdot (- 3 (\frac{3}{2})) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.8.2.1.1.1.3

重写表达式。

$1 - 3 \cdot 3 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 3 \cdot 3 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.8.2.1.1.2

将 $- 3$ 乘以 $3$ 。

$1 - 9 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 9 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.8.2.1.2

从 $1$ 中减去 $9$ 。

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.9

将所有不包含 $u_{33}$ 的项移到等式右边。

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解题步骤 3.2.9.1

在等式两边都加上 $8$ 。

$u_{33} = - 5 + 8$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.9.2

将 $- 5$ 和 $8$ 相加。

$u_{33} = 3$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$u_{33} = 3$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

解题步骤 3.2.10

求解方程组。

$u_{33} = 3 l_{32} = \frac{3}{2} u_{23} = - 6 u_{22} = - 2 l_{31} = 1 l_{21} = 3 u_{11} = 1 u_{12} = 1 u_{13} = 1$

解题步骤 3.2.11

列出所有解。

$u_{33} = 3, l_{32} = \frac{3}{2}, u_{23} = - 6, u_{22} = - 2, l_{31} = 1, l_{21} = 3, u_{11} = 1, u_{12} = 1, u_{13} = 1$

解题步骤 4

代入已求解的值。

$[\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}] = [\begin{array}{c} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 1 & \frac{3}{2} & 1 \end{array}] [\begin{array}{c} 1 & 1 & 1 \\ 0 & - 2 & - 6 \\ 0 & 0 & 3 \end{array}]$

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