数学
         
语言:中文    Language:English
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。
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$$\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}$$

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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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