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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\ &1\ &6\ \\ &3\ &5\ &7\ \\ &4\ &9\ &2\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &8\ &1\ &6\ \\ &3\ &5\ &7\ \\ &4\ &9\ &2\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {cccc | ccc} &8\ &1\ &6\ &1\ &0\ &0\ \\ &3\ &5\ &7\ &0\ &1\ &0\ \\ &4\ &9\ &2\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {cccc | ccc} &8\ &1\ &6\ &1\ &0\ &0\ \\ &0\ &\frac{37}{8}\ &\frac{19}{4}\ &-\frac{3}{8}\ &1\ &0\ \\ &0\ &\frac{17}{2}\ &-1\ &-\frac{1}{2}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccc | ccc} &8\ &1\ &6\ &1\ &0\ &0\ \\ &0\ &\frac{37}{8}\ &\frac{19}{4}\ &-\frac{3}{8}\ &1\ &0\ \\ &0\ &0\ &-\frac{360}{37}\ &\frac{7}{37}\ &-\frac{68}{37}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ above\ the\ diagonal\ to\ 0}\\\\->\ \ &\left (\begin{array} {cccc | ccc} &8\ &1\ &0\ &\frac{67}{60}\ &-\frac{17}{15}\ &\frac{37}{60}\ \\ &0\ &\frac{37}{8}\ &0\ &-\frac{407}{1440}\ &\frac{37}{360}\ &\frac{703}{1440}\ \\ &0\ &0\ &-\frac{360}{37}\ &\frac{7}{37}\ &-\frac{68}{37}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccc | ccc} &8\ &0\ &0\ &\frac{53}{45}\ &-\frac{52}{45}\ &\frac{23}{45}\ \\ &0\ &\frac{37}{8}\ &0\ &-\frac{407}{1440}\ &\frac{37}{360}\ &\frac{703}{1440}\ \\ &0\ &0\ &-\frac{360}{37}\ &\frac{7}{37}\ &-\frac{68}{37}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ on\ the\ main\ diagonal\ to\ 1}\\\\->\ \ &\left (\begin{array} {cccc | ccc} &1\ &0\ &0\ &\frac{53}{360}\ &-\frac{13}{90}\ &\frac{23}{360}\ \\ &0\ &\frac{37}{8}\ &0\ &-\frac{407}{1440}\ &\frac{37}{360}\ &\frac{703}{1440}\ \\ &0\ &0\ &-\frac{360}{37}\ &\frac{7}{37}\ &-\frac{68}{37}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccc | ccc} &1\ &0\ &0\ &\frac{53}{360}\ &-\frac{13}{90}\ &\frac{23}{360}\ \\ &0\ &1\ &0\ &-\frac{11}{180}\ &\frac{1}{45}\ &\frac{19}{180}\ \\ &0\ &0\ &-\frac{360}{37}\ &\frac{7}{37}\ &-\frac{68}{37}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccc | ccc} &1\ &0\ &0\ &\frac{53}{360}\ &-\frac{13}{90}\ &\frac{23}{360}\ \\ &0\ &1\ &0\ &-\frac{11}{180}\ &\frac{1}{45}\ &\frac{19}{180}\ \\ &0\ &0\ &1\ &-\frac{7}{360}\ &\frac{17}{90}\ &-\frac{37}{360}\ \\\end{array} \right )\\\\&\color{grey}{The\ inverse\ matrix\ obtained\ is\ : }\\&\begin{pmatrix} &\frac{53}{360}\ &-\frac{13}{90}\ &\frac{23}{360}\ \\ &-\frac{11}{180}\ &\frac{1}{45}\ &\frac{19}{180}\ \\ &-\frac{7}{360}\ &\frac{17}{90}\ &-\frac{37}{360}\ \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.