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Get the inverse matrix:
Enter an invertible matrix, with each element separated by a comma and each row ending with a semicolon.
Note that mathematical functions and variables are not supported.
Enter an invertible matrix, with each element separated by a comma and each row ending with a semicolon.
Note that mathematical functions and variables are not supported.
Current location：Linear algebra >Inverse matrix >History of inverse matrices >Answer
$$\begin{aligned}&\\ \color{black}{Calcu}&\color{black}{late\ the\ inverse\ matrix\ of\ } \ \ \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}{\ .}\\ \\Solu&tion:\\ &\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}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\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}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\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}{Convert\ elements\ above\ the\ diagonal\ to\ 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}{Convert\ elements\ on\ the\ main\ diagonal\ to\ 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}{The\ inverse\ matrix\ obtained\ is\ : }\\&\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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Elementary transformations of matrices:
Definition:Applying the following three transformations to the rows (columns) of a matrix becomes the elementary transformation of the matrix：
(1) Swap the positions of two rows (columns) in a matrix;
(2) Using non-zero constants λ Multiply a certain row (column) of a matrix;
(3) Convert a row (column) of a matrix γ Multiply to another row (column) of the matrix.