行列式与代数余子式

从最简单的行列式求解

\left|\begin{array}{c} a & b \\c & d \end{array} \right| =\left|\begin{array}{c} a & 0 \\c &a好看的法国电影mp; d \end{array} \right| + \left|\begin{array}{c} 0 & b \\c & d \end{array} \right| （根据性质3b)\\ =\left|\begin{array}{c} a & 0 \\c & 0 \end{array} \right| + \left|\begin{array}{c} a & 0 \\0 & d \end{array} \right| +\left|\begin{ar9位qqray}{c} 0 & b \\0 & d \end{array} \right| + \left|\begin{array}{c} 0 & b \\c & 0基金季报 \end{arra淘宝客是什么y} \right| =ad-bc

三阶行列式以同样的方式化解：

\left|\begin{array}{c} a_{11} & a_{12} &a_{13} \\a_{红米note521} & a_{22} &a_{23}\\a_{31} & a_{32} &a_{33} \end{array} \right| =\left|\begin{array}{c} a_{11关山月作者} & 0 &0 \\0 & a_{22} &0\\0 &a一个鸡蛋的暴走mp; 0 &a_{33} \end{array} \right| +\left|\begin{array}{c} a_{11} & 0 &0 \\0 & 0 &a_{23}\\0 & a_{32} &0征友 \end{array} \right| +\left|\begin{array}{c} 0 & a_{12} &0 \\a_{21} & 0 &0\\0 & 0 &a_{引言写什么33} \end{array} \right| \\+\left|\begin{array}{c}调度系统 0 & 小wa_{12} &0 \\0 & 0 &a_{23}\\a_{31} & 0 &0 \end{array} \right| +\left|\begin{array}{c} 0 & 0 &a_{13} \\a_{21} & 0 &0\\0 & a_{32} &0 \end{array} \right| +\left|\begin{array}{c} 0 & 0 &a_{13} \\0 & a_{22} &0\\a_{31} & 0 &0 \end{array} \right| \\ =a_{11}a_{12}a_{13} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31} + a_{13}a_{21}a_{3脱发能治好吗2}第七交响曲-a_{13}a_{22}a_{31}

对于一个n阶方阵，得到一个通用的行列式计算公式 det(A) = \sum_{n!} \pm a_{1\alpha}a_{2\beta}a_{3\gamma} ...a_{n\omega} \quad\quad (\alpha,\beta,\gamma,\omega) = {1,2,3 ....n}

\left|\begin{array}{c} a_{11} & a_{12} &a_怎么交朋友{13} \\a_{21} & a_{22} &a_{23}\\a_{31} & a_{32} &a_{33} \end{array} \right| =a_{11}a_{12}a_{13} - a_{11}a_{23}a_{32林姓男孩名字大全} - a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}\\ =a_{11}(a_{12}a_{13} - a_{23}a_{32}) + a_{12}(-a_{21}a_{33}+a_{23}a_{31}) + a_{13}(a_{21}a_{32}-a_{22}a_{31})

则 a_{11} 的代数余子式，就是 a_{12}a_{13} - a_{23}a_{32} ，化成行列式的形式： \left|\begin{array}{c} a_{22} &a_{23}\\a_{32} &a_{33} \end{array} \right| （去除 a_{11} 所在的行和意大利狐狸犬列，剩余的部分就是其代数余子式）

符号为当前 a_{ij} 所确性爱描述定，i+j=偶数为正，=奇数为负

公式： 蒙蒂塞洛水坝det(A) = a_{i1}C_{i1}+a_{i2}C_{i2} + ... + a_{in}C_{in} \quad \quad i=1或者其它

举例：

\left|\begin{array}{c} 1 & 2 & 0 \\2 & 3 &4\\5 & 6 &婴儿补钙amp;7 \end{array} \right| = 1\times \left|\begin{array}{c} 3 &4\\6 &7 \end{array} \right| + 2 \times (-1)\left|\begin{array}{c} 2 &4\\5 &7 \end{array} \right| + 0 \times \left|\begin{array}{c} 2 &3\\5 &6 \end{array} \right| = 9