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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} &42\ &-9\ &-19\ &15\ &-5\ \\ &-9\ &3\ &4\ &-4\ &1\ \\ &-19\ &4\ &7\ &-5\ &2\ \\ &15\ &-4\ &-5\ &5\ &-2\ \\ &-5\ &1\ &2\ &-2\ &1\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &42\ &-9\ &-19\ &15\ &-5\ \\ &-9\ &3\ &4\ &-4\ &1\ \\ &-19\ &4\ &7\ &-5\ &2\ \\ &15\ &-4\ &-5\ &5\ &-2\ \\ &-5\ &1\ &2\ &-2\ &1\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {cccccc | ccccc} &42\ &-9\ &-19\ &15\ &-5\ &1\ &0\ &0\ &0\ &0\ \\ &-9\ &3\ &4\ &-4\ &1\ &0\ &1\ &0\ &0\ &0\ \\ &-19\ &4\ &7\ &-5\ &2\ &0\ &0\ &1\ &0\ &0\ \\ &15\ &-4\ &-5\ &5\ &-2\ &0\ &0\ &0\ &1\ &0\ \\ &-5\ &1\ &2\ &-2\ &1\ &0\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &42\ &-9\ &-19\ &15\ &-5\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{15}{14}\ &-\frac{1}{14}\ &-\frac{11}{14}\ &-\frac{1}{14}\ &\frac{3}{14}\ &1\ &0\ &0\ &0\ \\ &0\ &-\frac{1}{14}\ &-\frac{67}{42}\ &\frac{25}{14}\ &-\frac{11}{42}\ &\frac{19}{42}\ &0\ &1\ &0\ &0\ \\ &0\ &-\frac{11}{14}\ &\frac{25}{14}\ &-\frac{5}{14}\ &-\frac{3}{14}\ &-\frac{5}{14}\ &0\ &0\ &1\ &0\ \\ &0\ &-\frac{1}{14}\ &-\frac{11}{42}\ &-\frac{3}{14}\ &\frac{17}{42}\ &\frac{5}{42}\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &42\ &-9\ &-19\ &15\ &-5\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{15}{14}\ &-\frac{1}{14}\ &-\frac{11}{14}\ &-\frac{1}{14}\ &\frac{3}{14}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &-\frac{8}{5}\ &\frac{26}{15}\ &-\frac{4}{15}\ &\frac{7}{15}\ &\frac{1}{15}\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{26}{15}\ &-\frac{14}{15}\ &-\frac{4}{15}\ &-\frac{1}{5}\ &\frac{11}{15}\ &0\ &1\ &0\ \\ &0\ &0\ &-\frac{4}{15}\ &-\frac{4}{15}\ &\frac{2}{5}\ &\frac{2}{15}\ &\frac{1}{15}\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &42\ &-9\ &-19\ &15\ &-5\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{15}{14}\ &-\frac{1}{14}\ &-\frac{11}{14}\ &-\frac{1}{14}\ &\frac{3}{14}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &-\frac{8}{5}\ &\frac{26}{15}\ &-\frac{4}{15}\ &\frac{7}{15}\ &\frac{1}{15}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &-\frac{5}{9}\ &\frac{11}{36}\ &\frac{29}{36}\ &\frac{13}{12}\ &1\ &0\ \\ &0\ &0\ &0\ &-\frac{5}{9}\ &\frac{4}{9}\ &\frac{1}{18}\ &\frac{1}{18}\ &-\frac{1}{6}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &42\ &-9\ &-19\ &15\ &-5\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{15}{14}\ &-\frac{1}{14}\ &-\frac{11}{14}\ &-\frac{1}{14}\ &\frac{3}{14}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &-\frac{8}{5}\ &\frac{26}{15}\ &-\frac{4}{15}\ &\frac{7}{15}\ &\frac{1}{15}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &-\frac{5}{9}\ &\frac{11}{36}\ &\frac{29}{36}\ &\frac{13}{12}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ above\ the\ diagonal\ to\ 0}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &42\ &-9\ &-19\ &15\ &0\ &11\ &\frac{45}{2}\ &20\ &25\ &\frac{85}{2}\ \\ &0\ &\frac{15}{14}\ &-\frac{1}{14}\ &-\frac{11}{14}\ &0\ &\frac{5}{14}\ &\frac{37}{28}\ &\frac{2}{7}\ &\frac{5}{14}\ &\frac{17}{28}\ \\ &0\ &0\ &-\frac{8}{5}\ &\frac{26}{15}\ &0\ &1\ &\frac{19}{15}\ &\frac{31}{15}\ &\frac{4}{3}\ &\frac{34}{15}\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &0\ &\frac{17}{12}\ &\frac{119}{36}\ &\frac{119}{36}\ &\frac{34}{9}\ &\frac{85}{18}\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &42\ &-9\ &-19\ &0\ &0\ &-\frac{23}{2}\ &-30\ &-\frac{65}{2}\ &-35\ &-\frac{65}{2}\ \\ &0\ &\frac{15}{14}\ &-\frac{1}{14}\ &0\ &0\ &\frac{43}{28}\ &\frac{57}{14}\ &\frac{85}{28}\ &\frac{7}{2}\ &\frac{127}{28}\ \\ &0\ &0\ &-\frac{8}{5}\ &0\ &0\ &-\frac{8}{5}\ &-\frac{24}{5}\ &-4\ &-\frac{28}{5}\ &-\frac{32}{5}\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &0\ &\frac{17}{12}\ &\frac{119}{36}\ &\frac{119}{36}\ &\frac{34}{9}\ &\frac{85}{18}\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &42\ &-9\ &0\ &0\ &0\ &\frac{15}{2}\ &27\ &15\ &\frac{63}{2}\ &\frac{87}{2}\ \\ &0\ &\frac{15}{14}\ &0\ &0\ &0\ &\frac{45}{28}\ &\frac{30}{7}\ &\frac{45}{14}\ &\frac{15}{4}\ &\frac{135}{28}\ \\ &0\ &0\ &-\frac{8}{5}\ &0\ &0\ &-\frac{8}{5}\ &-\frac{24}{5}\ &-4\ &-\frac{28}{5}\ &-\frac{32}{5}\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &0\ &\frac{17}{12}\ &\frac{119}{36}\ &\frac{119}{36}\ &\frac{34}{9}\ &\frac{85}{18}\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &42\ &0\ &0\ &0\ &0\ &21\ &63\ &42\ &63\ &84\ \\ &0\ &\frac{15}{14}\ &0\ &0\ &0\ &\frac{45}{28}\ &\frac{30}{7}\ &\frac{45}{14}\ &\frac{15}{4}\ &\frac{135}{28}\ \\ &0\ &0\ &-\frac{8}{5}\ &0\ &0\ &-\frac{8}{5}\ &-\frac{24}{5}\ &-4\ &-\frac{28}{5}\ &-\frac{32}{5}\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &0\ &\frac{17}{12}\ &\frac{119}{36}\ &\frac{119}{36}\ &\frac{34}{9}\ &\frac{85}{18}\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &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{1}{2}\ &\frac{3}{2}\ &1\ &\frac{3}{2}\ &2\ \\ &0\ &\frac{15}{14}\ &0\ &0\ &0\ &\frac{45}{28}\ &\frac{30}{7}\ &\frac{45}{14}\ &\frac{15}{4}\ &\frac{135}{28}\ \\ &0\ &0\ &-\frac{8}{5}\ &0\ &0\ &-\frac{8}{5}\ &-\frac{24}{5}\ &-4\ &-\frac{28}{5}\ &-\frac{32}{5}\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &0\ &\frac{17}{12}\ &\frac{119}{36}\ &\frac{119}{36}\ &\frac{34}{9}\ &\frac{85}{18}\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{1}{2}\ &\frac{3}{2}\ &1\ &\frac{3}{2}\ &2\ \\ &0\ &1\ &0\ &0\ &0\ &\frac{3}{2}\ &4\ &3\ &\frac{7}{2}\ &\frac{9}{2}\ \\ &0\ &0\ &-\frac{8}{5}\ &0\ &0\ &-\frac{8}{5}\ &-\frac{24}{5}\ &-4\ &-\frac{28}{5}\ &-\frac{32}{5}\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &0\ &\frac{17}{12}\ &\frac{119}{36}\ &\frac{119}{36}\ &\frac{34}{9}\ &\frac{85}{18}\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{1}{2}\ &\frac{3}{2}\ &1\ &\frac{3}{2}\ &2\ \\ &0\ &1\ &0\ &0\ &0\ &\frac{3}{2}\ &4\ &3\ &\frac{7}{2}\ &\frac{9}{2}\ \\ &0\ &0\ &1\ &0\ &0\ &1\ &3\ &\frac{5}{2}\ &\frac{7}{2}\ &4\ \\ &0\ &0\ &0\ &\frac{17}{18}\ &0\ &\frac{17}{12}\ &\frac{119}{36}\ &\frac{119}{36}\ &\frac{34}{9}\ &\frac{85}{18}\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{1}{2}\ &\frac{3}{2}\ &1\ &\frac{3}{2}\ &2\ \\ &0\ &1\ &0\ &0\ &0\ &\frac{3}{2}\ &4\ &3\ &\frac{7}{2}\ &\frac{9}{2}\ \\ &0\ &0\ &1\ &0\ &0\ &1\ &3\ &\frac{5}{2}\ &\frac{7}{2}\ &4\ \\ &0\ &0\ &0\ &1\ &0\ &\frac{3}{2}\ &\frac{7}{2}\ &\frac{7}{2}\ &4\ &5\ \\ &0\ &0\ &0\ &0\ &\frac{2}{17}\ &\frac{4}{17}\ &\frac{9}{17}\ &\frac{8}{17}\ &\frac{10}{17}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &\frac{1}{2}\ &\frac{3}{2}\ &1\ &\frac{3}{2}\ &2\ \\ &0\ &1\ &0\ &0\ &0\ &\frac{3}{2}\ &4\ &3\ &\frac{7}{2}\ &\frac{9}{2}\ \\ &0\ &0\ &1\ &0\ &0\ &1\ &3\ &\frac{5}{2}\ &\frac{7}{2}\ &4\ \\ &0\ &0\ &0\ &1\ &0\ &\frac{3}{2}\ &\frac{7}{2}\ &\frac{7}{2}\ &4\ &5\ \\ &0\ &0\ &0\ &0\ &1\ &2\ &\frac{9}{2}\ &4\ &5\ &\frac{17}{2}\ \\\end{array} \right )\\\\&\color{grey}{The\ inverse\ matrix\ obtained\ is\ : }\\&\begin{pmatrix} &\frac{1}{2}\ &\frac{3}{2}\ &1\ &\frac{3}{2}\ &2\ \\ &\frac{3}{2}\ &4\ &3\ &\frac{7}{2}\ &\frac{9}{2}\ \\ &1\ &3\ &\frac{5}{2}\ &\frac{7}{2}\ &4\ \\ &\frac{3}{2}\ &\frac{7}{2}\ &\frac{7}{2}\ &4\ &5\ \\ &2\ &\frac{9}{2}\ &4\ &5\ &\frac{17}{2}\ \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.