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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} &1\ &2\ &7\ &8\ \\ &5\ &4\ &9\ &3\ \\ &6\ &6\ &7\ &7\ \\ &2\ &8\ &5\ &4\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &1\ &2\ &7\ &8\ \\ &5\ &4\ &9\ &3\ \\ &6\ &6\ &7\ &7\ \\ &2\ &8\ &5\ &4\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {ccccc | cccc} &1\ &2\ &7\ &8\ &1\ &0\ &0\ &0\ \\ &5\ &4\ &9\ &3\ &0\ &1\ &0\ &0\ \\ &6\ &6\ &7\ &7\ &0\ &0\ &1\ &0\ \\ &2\ &8\ &5\ &4\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &2\ &7\ &8\ &1\ &0\ &0\ &0\ \\ &0\ &-6\ &-26\ &-37\ &-5\ &1\ &0\ &0\ \\ &0\ &-6\ &-35\ &-41\ &-6\ &0\ &1\ &0\ \\ &0\ &4\ &-9\ &-12\ &-2\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &2\ &7\ &8\ &1\ &0\ &0\ &0\ \\ &0\ &-6\ &-26\ &-37\ &-5\ &1\ &0\ &0\ \\ &0\ &0\ &-9\ &-4\ &-1\ &-1\ &1\ &0\ \\ &0\ &0\ &-\frac{79}{3}\ &-\frac{110}{3}\ &-\frac{16}{3}\ &\frac{2}{3}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &2\ &7\ &8\ &1\ &0\ &0\ &0\ \\ &0\ &-6\ &-26\ &-37\ &-5\ &1\ &0\ &0\ \\ &0\ &0\ &-9\ &-4\ &-1\ &-1\ &1\ &0\ \\ &0\ &0\ &0\ &-\frac{674}{27}\ &-\frac{65}{27}\ &\frac{97}{27}\ &-\frac{79}{27}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ above\ the\ diagonal\ to\ 0}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &2\ &7\ &0\ &\frac{77}{337}\ &\frac{388}{337}\ &-\frac{316}{337}\ &\frac{108}{337}\ \\ &0\ &-6\ &-26\ &0\ &-\frac{965}{674}\ &-\frac{2915}{674}\ &\frac{2923}{674}\ &-\frac{999}{674}\ \\ &0\ &0\ &-9\ &0\ &-\frac{207}{337}\ &-\frac{531}{337}\ &\frac{495}{337}\ &-\frac{54}{337}\ \\ &0\ &0\ &0\ &-\frac{674}{27}\ &-\frac{65}{27}\ &\frac{97}{27}\ &-\frac{79}{27}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &2\ &0\ &0\ &-\frac{84}{337}\ &-\frac{25}{337}\ &\frac{69}{337}\ &\frac{66}{337}\ \\ &0\ &-6\ &0\ &0\ &\frac{77847}{227138}\ &\frac{51561}{227138}\ &\frac{21231}{227138}\ &-\frac{687}{674}\ \\ &0\ &0\ &-9\ &0\ &-\frac{207}{337}\ &-\frac{531}{337}\ &\frac{495}{337}\ &-\frac{54}{337}\ \\ &0\ &0\ &0\ &-\frac{674}{27}\ &-\frac{65}{27}\ &\frac{97}{27}\ &-\frac{79}{27}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &-\frac{30667}{227138}\ &\frac{1}{674}\ &\frac{53583}{227138}\ &-\frac{32689}{227138}\ \\ &0\ &-6\ &0\ &0\ &\frac{77847}{227138}\ &\frac{51561}{227138}\ &\frac{21231}{227138}\ &-\frac{687}{674}\ \\ &0\ &0\ &-9\ &0\ &-\frac{207}{337}\ &-\frac{531}{337}\ &\frac{495}{337}\ &-\frac{54}{337}\ \\ &0\ &0\ &0\ &-\frac{674}{27}\ &-\frac{65}{27}\ &\frac{97}{27}\ &-\frac{79}{27}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ on\ the\ main\ diagonal\ to\ 1}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &-\frac{30667}{227138}\ &\frac{1}{674}\ &\frac{53583}{227138}\ &-\frac{32689}{227138}\ \\ &0\ &1\ &0\ &0\ &-\frac{25949}{454276}\ &-\frac{17187}{454276}\ &-\frac{7077}{454276}\ &\frac{229}{1348}\ \\ &0\ &0\ &-9\ &0\ &-\frac{207}{337}\ &-\frac{531}{337}\ &\frac{495}{337}\ &-\frac{54}{337}\ \\ &0\ &0\ &0\ &-\frac{674}{27}\ &-\frac{65}{27}\ &\frac{97}{27}\ &-\frac{79}{27}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &-\frac{30667}{227138}\ &\frac{1}{674}\ &\frac{53583}{227138}\ &-\frac{32689}{227138}\ \\ &0\ &1\ &0\ &0\ &-\frac{25949}{454276}\ &-\frac{17187}{454276}\ &-\frac{7077}{454276}\ &\frac{229}{1348}\ \\ &0\ &0\ &1\ &0\ &\frac{23}{337}\ &\frac{59}{337}\ &-\frac{55}{337}\ &\frac{6}{337}\ \\ &0\ &0\ &0\ &-\frac{674}{27}\ &-\frac{65}{27}\ &\frac{97}{27}\ &-\frac{79}{27}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &-\frac{30667}{227138}\ &\frac{1}{674}\ &\frac{53583}{227138}\ &-\frac{32689}{227138}\ \\ &0\ &1\ &0\ &0\ &-\frac{25949}{454276}\ &-\frac{17187}{454276}\ &-\frac{7077}{454276}\ &\frac{229}{1348}\ \\ &0\ &0\ &1\ &0\ &\frac{23}{337}\ &\frac{59}{337}\ &-\frac{55}{337}\ &\frac{6}{337}\ \\ &0\ &0\ &0\ &1\ &\frac{65}{674}\ &-\frac{97}{674}\ &\frac{79}{674}\ &-\frac{27}{674}\ \\\end{array} \right )\\\\&\color{grey}{The\ inverse\ matrix\ obtained\ is\ : }\\&\begin{pmatrix} &-\frac{30667}{227138}\ &\frac{1}{674}\ &\frac{53583}{227138}\ &-\frac{32689}{227138}\ \\ &-\frac{25949}{454276}\ &-\frac{17187}{454276}\ &-\frac{7077}{454276}\ &\frac{229}{1348}\ \\ &\frac{23}{337}\ &\frac{59}{337}\ &-\frac{55}{337}\ &\frac{6}{337}\ \\ &\frac{65}{674}\ &-\frac{97}{674}\ &\frac{79}{674}\ &-\frac{27}{674}\ \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.