求逆矩阵：
输入一个可逆矩阵，每个元用逗号隔开，每行用分号结尾。
注意，不支持支持数学函数和变量。
输入一个可逆矩阵，每个元用逗号隔开，每行用分号结尾。
注意，不支持支持数学函数和变量。
$$\begin{aligned}&\\ \color{black}{计算矩阵}& \ \ \begin{pmatrix} &3\ &2\ &2\ &2\ &2\ \\ &2\ &3\ &2\ &2\ &2\ \\ &2\ &2\ &3\ &2\ &2\ \\ &2\ &2\ &2\ &3\ &2\ \\ &2\ &2\ &2\ &2\ &3\ \end{pmatrix}\color{black}{的逆矩阵。}\\ \\解:&\\ &\begin{pmatrix} &3\ &2\ &2\ &2\ &2\ \\ &2\ &3\ &2\ &2\ &2\ \\ &2\ &2\ &3\ &2\ &2\ \\ &2\ &2\ &2\ &3\ &2\ \\ &2\ &2\ &2\ &2\ &3\ \end{pmatrix}\\\\&\color{grey}{用矩阵的初等变换来求逆矩阵：}\\&\left (\begin{array} {cccccc | ccccc} &3\ &2\ &2\ &2\ &2\ &1\ &0\ &0\ &0\ &0\ \\ &2\ &3\ &2\ &2\ &2\ &0\ &1\ &0\ &0\ &0\ \\ &2\ &2\ &3\ &2\ &2\ &0\ &0\ &1\ &0\ &0\ \\ &2\ &2\ &2\ &3\ &2\ &0\ &0\ &0\ &1\ &0\ \\ &2\ &2\ &2\ &2\ &3\ &0\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{将已知矩阵化为上三角矩阵}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &3\ &2\ &2\ &2\ &2\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{5}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &-\frac{2}{3}\ &1\ &0\ &0\ &0\ \\ &0\ &\frac{2}{3}\ &\frac{5}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &-\frac{2}{3}\ &0\ &1\ &0\ &0\ \\ &0\ &\frac{2}{3}\ &\frac{2}{3}\ &\frac{5}{3}\ &\frac{2}{3}\ &-\frac{2}{3}\ &0\ &0\ &1\ &0\ \\ &0\ &\frac{2}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &\frac{5}{3}\ &-\frac{2}{3}\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &3\ &2\ &2\ &2\ &2\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{5}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &-\frac{2}{3}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{7}{5}\ &\frac{2}{5}\ &\frac{2}{5}\ &-\frac{2}{5}\ &-\frac{2}{5}\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{2}{5}\ &\frac{7}{5}\ &\frac{2}{5}\ &-\frac{2}{5}\ &-\frac{2}{5}\ &0\ &1\ &0\ \\ &0\ &0\ &\frac{2}{5}\ &\frac{2}{5}\ &\frac{7}{5}\ &-\frac{2}{5}\ &-\frac{2}{5}\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &3\ &2\ &2\ &2\ &2\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{5}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &-\frac{2}{3}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{7}{5}\ &\frac{2}{5}\ &\frac{2}{5}\ &-\frac{2}{5}\ &-\frac{2}{5}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &\frac{2}{7}\ &-\frac{2}{7}\ &-\frac{2}{7}\ &-\frac{2}{7}\ &1\ &0\ \\ &0\ &0\ &0\ &\frac{2}{7}\ &\frac{9}{7}\ &-\frac{2}{7}\ &-\frac{2}{7}\ &-\frac{2}{7}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &3\ &2\ &2\ &2\ &2\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{5}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &-\frac{2}{3}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{7}{5}\ &\frac{2}{5}\ &\frac{2}{5}\ &-\frac{2}{5}\ &-\frac{2}{5}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &\frac{2}{7}\ &-\frac{2}{7}\ &-\frac{2}{7}\ &-\frac{2}{7}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\&\color{grey}{将对角线以上的元素化为0}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &3\ &2\ &2\ &2\ &0\ &\frac{15}{11}\ &\frac{4}{11}\ &\frac{4}{11}\ &\frac{4}{11}\ &-\frac{18}{11}\ \\ &0\ &\frac{5}{3}\ &\frac{2}{3}\ &\frac{2}{3}\ &0\ &-\frac{6}{11}\ &\frac{37}{33}\ &\frac{4}{33}\ &\frac{4}{33}\ &-\frac{6}{11}\ \\ &0\ &0\ &\frac{7}{5}\ &\frac{2}{5}\ &0\ &-\frac{18}{55}\ &-\frac{18}{55}\ &\frac{59}{55}\ &\frac{4}{55}\ &-\frac{18}{55}\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &0\ &-\frac{18}{77}\ &-\frac{18}{77}\ &-\frac{18}{77}\ &\frac{81}{77}\ &-\frac{18}{77}\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &3\ &2\ &2\ &0\ &0\ &\frac{19}{11}\ &\frac{8}{11}\ &\frac{8}{11}\ &-\frac{14}{11}\ &-\frac{14}{11}\ \\ &0\ &\frac{5}{3}\ &\frac{2}{3}\ &0\ &0\ &-\frac{14}{33}\ &\frac{41}{33}\ &\frac{8}{33}\ &-\frac{14}{33}\ &-\frac{14}{33}\ \\ &0\ &0\ &\frac{7}{5}\ &0\ &0\ &-\frac{14}{55}\ &-\frac{14}{55}\ &\frac{63}{55}\ &-\frac{14}{55}\ &-\frac{14}{55}\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &0\ &-\frac{18}{77}\ &-\frac{18}{77}\ &-\frac{18}{77}\ &\frac{81}{77}\ &-\frac{18}{77}\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &3\ &2\ &0\ &0\ &0\ &\frac{23}{11}\ &\frac{12}{11}\ &-\frac{10}{11}\ &-\frac{10}{11}\ &-\frac{10}{11}\ \\ &0\ &\frac{5}{3}\ &0\ &0\ &0\ &-\frac{10}{33}\ &\frac{15}{11}\ &-\frac{10}{33}\ &-\frac{10}{33}\ &-\frac{10}{33}\ \\ &0\ &0\ &\frac{7}{5}\ &0\ &0\ &-\frac{14}{55}\ &-\frac{14}{55}\ &\frac{63}{55}\ &-\frac{14}{55}\ &-\frac{14}{55}\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &0\ &-\frac{18}{77}\ &-\frac{18}{77}\ &-\frac{18}{77}\ &\frac{81}{77}\ &-\frac{18}{77}\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &3\ &0\ &0\ &0\ &0\ &\frac{27}{11}\ &-\frac{6}{11}\ &-\frac{6}{11}\ &-\frac{6}{11}\ &-\frac{6}{11}\ \\ &0\ &\frac{5}{3}\ &0\ &0\ &0\ &-\frac{10}{33}\ &\frac{15}{11}\ &-\frac{10}{33}\ &-\frac{10}{33}\ &-\frac{10}{33}\ \\ &0\ &0\ &\frac{7}{5}\ &0\ &0\ &-\frac{14}{55}\ &-\frac{14}{55}\ &\frac{63}{55}\ &-\frac{14}{55}\ &-\frac{14}{55}\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &0\ &-\frac{18}{77}\ &-\frac{18}{77}\ &-\frac{18}{77}\ &\frac{81}{77}\ &-\frac{18}{77}\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\&\color{grey}{将主对角线元素化为1}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &\frac{5}{3}\ &0\ &0\ &0\ &-\frac{10}{33}\ &\frac{15}{11}\ &-\frac{10}{33}\ &-\frac{10}{33}\ &-\frac{10}{33}\ \\ &0\ &0\ &\frac{7}{5}\ &0\ &0\ &-\frac{14}{55}\ &-\frac{14}{55}\ &\frac{63}{55}\ &-\frac{14}{55}\ &-\frac{14}{55}\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &0\ &-\frac{18}{77}\ &-\frac{18}{77}\ &-\frac{18}{77}\ &\frac{81}{77}\ &-\frac{18}{77}\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &1\ &0\ &0\ &0\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &\frac{7}{5}\ &0\ &0\ &-\frac{14}{55}\ &-\frac{14}{55}\ &\frac{63}{55}\ &-\frac{14}{55}\ &-\frac{14}{55}\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &0\ &-\frac{18}{77}\ &-\frac{18}{77}\ &-\frac{18}{77}\ &\frac{81}{77}\ &-\frac{18}{77}\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &1\ &0\ &0\ &0\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &0\ &\frac{9}{7}\ &0\ &-\frac{18}{77}\ &-\frac{18}{77}\ &-\frac{18}{77}\ &\frac{81}{77}\ &-\frac{18}{77}\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &1\ &0\ &0\ &0\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &0\ &1\ &0\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &0\ &0\ &\frac{11}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &1\ &0\ &0\ &0\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &0\ &1\ &0\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ \\ &0\ &0\ &0\ &0\ &1\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ \\\end{array} \right )\\\\&\color{grey}{所求的逆矩阵为：}\\&\begin{pmatrix} &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ \\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ &-\frac{2}{11}\ \\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &-\frac{2}{11}\ &\frac{9}{11}\ \end{pmatrix}\end{aligned}$$
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矩阵的初等变换：
定义：对矩阵的行（列）施行下列三种变换都成为矩阵的初等变换：
（1）互换矩阵两行（列）的位置；
（2）用非零常数λ乘矩阵的某行（列）；
（3）将矩阵某行（列）的γ倍加到矩阵的另一行（列）上。
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