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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
$$\begin{aligned}&\\ \color{black}{Calcu}&\color{black}{late\ the\ inverse\ matrix\ of\ } \ \ \begin{pmatrix} &8\ &11\ &14\ &1\ \\ &13\ &2\ &7\ &12\ \\ &3\ &16\ &9\ &6\ \\ &10\ &5\ &4\ &15\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &8\ &11\ &14\ &1\ \\ &13\ &2\ &7\ &12\ \\ &3\ &16\ &9\ &6\ \\ &10\ &5\ &4\ &15\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {ccccc | cccc} &8\ &11\ &14\ &1\ &1\ &0\ &0\ &0\ \\ &13\ &2\ &7\ &12\ &0\ &1\ &0\ &0\ \\ &3\ &16\ &9\ &6\ &0\ &0\ &1\ &0\ \\ &10\ &5\ &4\ &15\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &8\ &11\ &14\ &1\ &1\ &0\ &0\ &0\ \\ &0\ &-\frac{127}{8}\ &-\frac{63}{4}\ &\frac{83}{8}\ &-\frac{13}{8}\ &1\ &0\ &0\ \\ &0\ &\frac{95}{8}\ &\frac{15}{4}\ &\frac{45}{8}\ &-\frac{3}{8}\ &0\ &1\ &0\ \\ &0\ &-\frac{35}{4}\ &-\frac{27}{2}\ &\frac{55}{4}\ &-\frac{5}{4}\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &8\ &11\ &14\ &1\ &1\ &0\ &0\ &0\ \\ &0\ &-\frac{127}{8}\ &-\frac{63}{4}\ &\frac{83}{8}\ &-\frac{13}{8}\ &1\ &0\ &0\ \\ &0\ &0\ &-\frac{1020}{127}\ &\frac{1700}{127}\ &-\frac{202}{127}\ &\frac{95}{127}\ &1\ &0\ \\ &0\ &0\ &-\frac{612}{127}\ &\frac{1020}{127}\ &-\frac{45}{127}\ &-\frac{70}{127}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &8\ &11\ &14\ &1\ &1\ &0\ &0\ &0\ \\ &0\ &-\frac{127}{8}\ &-\frac{63}{4}\ &\frac{83}{8}\ &-\frac{13}{8}\ &1\ &0\ &0\ \\ &0\ &0\ &-\frac{1020}{127}\ &\frac{1700}{127}\ &-\frac{202}{127}\ &\frac{95}{127}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &\frac{3}{5}\ &-1\ &-\frac{3}{5}\ &1\ \\\end{array} \right )\\\ \ &\color{red}{This\ matrix\ is\ an\ irreversible\ matrix.}\end{aligned}$$
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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.