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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} &12\ &34\ &1\ &2\ &3\ &4\ \\ &56\ &78\ &5\ &6\ &7\ &8\ \\ &13\ &57\ &1\ &3\ &5\ &7\ \\ &24\ &68\ &2\ &4\ &6\ &8\ \\ &15\ &26\ &1\ &5\ &2\ &6\ \\ &37\ &48\ &3\ &7\ &4\ &8\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &12\ &34\ &1\ &2\ &3\ &4\ \\ &56\ &78\ &5\ &6\ &7\ &8\ \\ &13\ &57\ &1\ &3\ &5\ &7\ \\ &24\ &68\ &2\ &4\ &6\ &8\ \\ &15\ &26\ &1\ &5\ &2\ &6\ \\ &37\ &48\ &3\ &7\ &4\ &8\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {ccccccc | cccccc} &12\ &34\ &1\ &2\ &3\ &4\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &56\ &78\ &5\ &6\ &7\ &8\ &0\ &1\ &0\ &0\ &0\ &0\ \\ &13\ &57\ &1\ &3\ &5\ &7\ &0\ &0\ &1\ &0\ &0\ &0\ \\ &24\ &68\ &2\ &4\ &6\ &8\ &0\ &0\ &0\ &1\ &0\ &0\ \\ &15\ &26\ &1\ &5\ &2\ &6\ &0\ &0\ &0\ &0\ &1\ &0\ \\ &37\ &48\ &3\ &7\ &4\ &8\ &0\ &0\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &12\ &34\ &1\ &2\ &3\ &4\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &0\ &-\frac{242}{3}\ &\frac{1}{3}\ &-\frac{10}{3}\ &-7\ &-\frac{32}{3}\ &-\frac{14}{3}\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{121}{6}\ &-\frac{1}{12}\ &\frac{5}{6}\ &\frac{7}{4}\ &\frac{8}{3}\ &-\frac{13}{12}\ &0\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &0\ &0\ &0\ &0\ &-2\ &0\ &0\ &1\ &0\ &0\ \\ &0\ &-\frac{33}{2}\ &-\frac{1}{4}\ &\frac{5}{2}\ &-\frac{7}{4}\ &1\ &-\frac{5}{4}\ &0\ &0\ &0\ &1\ &0\ \\ &0\ &-\frac{341}{6}\ &-\frac{1}{12}\ &\frac{5}{6}\ &-\frac{21}{4}\ &-\frac{13}{3}\ &-\frac{37}{12}\ &0\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &12\ &34\ &1\ &2\ &3\ &4\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &0\ &-\frac{242}{3}\ &\frac{1}{3}\ &-\frac{10}{3}\ &-7\ &-\frac{32}{3}\ &-\frac{14}{3}\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &0\ &0\ &0\ &0\ &0\ &-\frac{9}{4}\ &\frac{1}{4}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &0\ &0\ &0\ &0\ &-2\ &0\ &0\ &1\ &0\ &0\ \\ &0\ &0\ &-\frac{7}{22}\ &\frac{35}{11}\ &-\frac{7}{22}\ &\frac{35}{11}\ &-\frac{13}{44}\ &-\frac{9}{44}\ &0\ &0\ &1\ &0\ \\ &0\ &0\ &-\frac{7}{22}\ &\frac{35}{11}\ &-\frac{7}{22}\ &\frac{35}{11}\ &\frac{9}{44}\ &-\frac{31}{44}\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &12\ &34\ &1\ &2\ &3\ &4\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &0\ &-\frac{242}{3}\ &\frac{1}{3}\ &-\frac{10}{3}\ &-7\ &-\frac{32}{3}\ &-\frac{14}{3}\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &0\ &-\frac{7}{22}\ &\frac{35}{11}\ &-\frac{7}{22}\ &\frac{35}{11}\ &-\frac{13}{44}\ &-\frac{9}{44}\ &0\ &0\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &0\ &0\ &-2\ &0\ &0\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &0\ &0\ &0\ &-\frac{9}{4}\ &\frac{1}{4}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &0\ &0\ &0\ &0\ &\frac{1}{2}\ &-\frac{1}{2}\ &0\ &0\ &-1\ &1\ \\\end{array} \right )\\\ \ &\color{red}{This\ matrix\ is\ an\ irreversible\ matrix.}\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.