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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} &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}{\ .}\\ \\Solu&tion:\\ &\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}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\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}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\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}{Convert\ elements\ above\ the\ diagonal\ to\ 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}{Convert\ elements\ on\ the\ main\ diagonal\ to\ 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}{The\ inverse\ matrix\ obtained\ is\ : }\\&\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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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.