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} &2\ &1\ &1\ &1\ &1\ \\ &1\ &2\ &1\ &1\ &1\ \\ &1\ &1\ &2\ &1\ &1\ \\ &1\ &1\ &1\ &2\ &1\ \\ &1\ &1\ &1\ &1\ &2\ \end{pmatrix}\color{black}{的逆矩阵。}\\ \\解:&\\ &\begin{pmatrix} &2\ &1\ &1\ &1\ &1\ \\ &1\ &2\ &1\ &1\ &1\ \\ &1\ &1\ &2\ &1\ &1\ \\ &1\ &1\ &1\ &2\ &1\ \\ &1\ &1\ &1\ &1\ &2\ \end{pmatrix}\\\\&\color{grey}{用矩阵的初等变换来求逆矩阵：}\\&\left (\begin{array} {cccccc | ccccc} &2\ &1\ &1\ &1\ &1\ &1\ &0\ &0\ &0\ &0\ \\ &1\ &2\ &1\ &1\ &1\ &0\ &1\ &0\ &0\ &0\ \\ &1\ &1\ &2\ &1\ &1\ &0\ &0\ &1\ &0\ &0\ \\ &1\ &1\ &1\ &2\ &1\ &0\ &0\ &0\ &1\ &0\ \\ &1\ &1\ &1\ &1\ &2\ &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} &2\ &1\ &1\ &1\ &1\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{3}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &-\frac{1}{2}\ &1\ &0\ &0\ &0\ \\ &0\ &\frac{1}{2}\ &\frac{3}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &-\frac{1}{2}\ &0\ &1\ &0\ &0\ \\ &0\ &\frac{1}{2}\ &\frac{1}{2}\ &\frac{3}{2}\ &\frac{1}{2}\ &-\frac{1}{2}\ &0\ &0\ &1\ &0\ \\ &0\ &\frac{1}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &\frac{3}{2}\ &-\frac{1}{2}\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &2\ &1\ &1\ &1\ &1\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{3}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &-\frac{1}{2}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{4}{3}\ &\frac{1}{3}\ &\frac{1}{3}\ &-\frac{1}{3}\ &-\frac{1}{3}\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{1}{3}\ &\frac{4}{3}\ &\frac{1}{3}\ &-\frac{1}{3}\ &-\frac{1}{3}\ &0\ &1\ &0\ \\ &0\ &0\ &\frac{1}{3}\ &\frac{1}{3}\ &\frac{4}{3}\ &-\frac{1}{3}\ &-\frac{1}{3}\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &2\ &1\ &1\ &1\ &1\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{3}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &-\frac{1}{2}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{4}{3}\ &\frac{1}{3}\ &\frac{1}{3}\ &-\frac{1}{3}\ &-\frac{1}{3}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &\frac{1}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &1\ &0\ \\ &0\ &0\ &0\ &\frac{1}{4}\ &\frac{5}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &2\ &1\ &1\ &1\ &1\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{3}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &-\frac{1}{2}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{4}{3}\ &\frac{1}{3}\ &\frac{1}{3}\ &-\frac{1}{3}\ &-\frac{1}{3}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &\frac{1}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &1\ \\\end{array} \right )\\\\&\color{grey}{将对角线以上的元素化为0}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &2\ &1\ &1\ &1\ &0\ &\frac{7}{6}\ &\frac{1}{6}\ &\frac{1}{6}\ &\frac{1}{6}\ &-\frac{5}{6}\ \\ &0\ &\frac{3}{2}\ &\frac{1}{2}\ &\frac{1}{2}\ &0\ &-\frac{5}{12}\ &\frac{13}{12}\ &\frac{1}{12}\ &\frac{1}{12}\ &-\frac{5}{12}\ \\ &0\ &0\ &\frac{4}{3}\ &\frac{1}{3}\ &0\ &-\frac{5}{18}\ &-\frac{5}{18}\ &\frac{19}{18}\ &\frac{1}{18}\ &-\frac{5}{18}\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &0\ &-\frac{5}{24}\ &-\frac{5}{24}\ &-\frac{5}{24}\ &\frac{25}{24}\ &-\frac{5}{24}\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &2\ &1\ &1\ &0\ &0\ &\frac{4}{3}\ &\frac{1}{3}\ &\frac{1}{3}\ &-\frac{2}{3}\ &-\frac{2}{3}\ \\ &0\ &\frac{3}{2}\ &\frac{1}{2}\ &0\ &0\ &-\frac{1}{3}\ &\frac{7}{6}\ &\frac{1}{6}\ &-\frac{1}{3}\ &-\frac{1}{3}\ \\ &0\ &0\ &\frac{4}{3}\ &0\ &0\ &-\frac{2}{9}\ &-\frac{2}{9}\ &\frac{10}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &0\ &-\frac{5}{24}\ &-\frac{5}{24}\ &-\frac{5}{24}\ &\frac{25}{24}\ &-\frac{5}{24}\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &2\ &1\ &0\ &0\ &0\ &\frac{3}{2}\ &\frac{1}{2}\ &-\frac{1}{2}\ &-\frac{1}{2}\ &-\frac{1}{2}\ \\ &0\ &\frac{3}{2}\ &0\ &0\ &0\ &-\frac{1}{4}\ &\frac{5}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ \\ &0\ &0\ &\frac{4}{3}\ &0\ &0\ &-\frac{2}{9}\ &-\frac{2}{9}\ &\frac{10}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &0\ &-\frac{5}{24}\ &-\frac{5}{24}\ &-\frac{5}{24}\ &\frac{25}{24}\ &-\frac{5}{24}\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &2\ &0\ &0\ &0\ &0\ &\frac{5}{3}\ &-\frac{1}{3}\ &-\frac{1}{3}\ &-\frac{1}{3}\ &-\frac{1}{3}\ \\ &0\ &\frac{3}{2}\ &0\ &0\ &0\ &-\frac{1}{4}\ &\frac{5}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ \\ &0\ &0\ &\frac{4}{3}\ &0\ &0\ &-\frac{2}{9}\ &-\frac{2}{9}\ &\frac{10}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &0\ &-\frac{5}{24}\ &-\frac{5}{24}\ &-\frac{5}{24}\ &\frac{25}{24}\ &-\frac{5}{24}\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &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{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &\frac{3}{2}\ &0\ &0\ &0\ &-\frac{1}{4}\ &\frac{5}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ &-\frac{1}{4}\ \\ &0\ &0\ &\frac{4}{3}\ &0\ &0\ &-\frac{2}{9}\ &-\frac{2}{9}\ &\frac{10}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &0\ &-\frac{5}{24}\ &-\frac{5}{24}\ &-\frac{5}{24}\ &\frac{25}{24}\ &-\frac{5}{24}\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &1\ &0\ &0\ &0\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &\frac{4}{3}\ &0\ &0\ &-\frac{2}{9}\ &-\frac{2}{9}\ &\frac{10}{9}\ &-\frac{2}{9}\ &-\frac{2}{9}\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &0\ &-\frac{5}{24}\ &-\frac{5}{24}\ &-\frac{5}{24}\ &\frac{25}{24}\ &-\frac{5}{24}\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &1\ &0\ &0\ &0\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &0\ &\frac{5}{4}\ &0\ &-\frac{5}{24}\ &-\frac{5}{24}\ &-\frac{5}{24}\ &\frac{25}{24}\ &-\frac{5}{24}\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &1\ &0\ &0\ &0\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &0\ &1\ &0\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &0\ &0\ &\frac{6}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &-\frac{1}{5}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &1\ &0\ &0\ &0\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &0\ &1\ &0\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ \\ &0\ &0\ &0\ &0\ &1\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ \\\end{array} \right )\\\\&\color{grey}{所求的逆矩阵为：}\\&\begin{pmatrix} &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ \\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ &-\frac{1}{6}\ \\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &-\frac{1}{6}\ &\frac{5}{6}\ \end{pmatrix}\end{aligned}$$
你的问题在这里没有得到解决？请到 热门难题 里面看看吧！
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。
新增加学习笔记(安卓版)百度网盘快速下载应用程序，欢迎使用。
新增加学习笔记(安卓版)本站下载应用程序，欢迎使用。
新增线性代数行列式的计算，欢迎使用。
数学计算和一元方程已经支持正割函数和余割函数，欢迎使用。
新增加贷款计算器模块（具体位置：数学运算 > 贷款计算器），欢迎使用。