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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} &1168\ &2324\ &4624\ &9224\ \\ &2284\ &4550\ &9004\ &17942\ \\ &9808\ &19580\ &38992\ &77864\ \\ &14208\ &28356\ &56448\ &112704\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &1168\ &2324\ &4624\ &9224\ \\ &2284\ &4550\ &9004\ &17942\ \\ &9808\ &19580\ &38992\ &77864\ \\ &14208\ &28356\ &56448\ &112704\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {ccccc | cccc} &1168\ &2324\ &4624\ &9224\ &1\ &0\ &0\ &0\ \\ &2284\ &4550\ &9004\ &17942\ &0\ &1\ &0\ &0\ \\ &9808\ &19580\ &38992\ &77864\ &0\ &0\ &1\ &0\ \\ &14208\ &28356\ &56448\ &112704\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1168\ &2324\ &4624\ &9224\ &1\ &0\ &0\ &0\ \\ &0\ &\frac{399}{73}\ &-\frac{2784}{73}\ &-\frac{6960}{73}\ &-\frac{571}{292}\ &1\ &0\ &0\ \\ &0\ &\frac{4728}{73}\ &\frac{11904}{73}\ &\frac{29760}{73}\ &-\frac{613}{73}\ &0\ &1\ &0\ \\ &0\ &\frac{6276}{73}\ &\frac{14592}{73}\ &\frac{36480}{73}\ &-\frac{888}{73}\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1168\ &2324\ &4624\ &9224\ &1\ &0\ &0\ &0\ \\ &0\ &\frac{399}{73}\ &-\frac{2784}{73}\ &-\frac{6960}{73}\ &-\frac{571}{292}\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{81792}{133}\ &\frac{204480}{133}\ &\frac{1965}{133}\ &-\frac{1576}{133}\ &1\ &0\ \\ &0\ &0\ &\frac{106368}{133}\ &\frac{265920}{133}\ &\frac{2473}{133}\ &-\frac{2092}{133}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1168\ &2324\ &4624\ &9224\ &1\ &0\ &0\ &0\ \\ &0\ &\frac{399}{73}\ &-\frac{2784}{73}\ &-\frac{6960}{73}\ &-\frac{571}{292}\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{81792}{133}\ &\frac{204480}{133}\ &\frac{1965}{133}\ &-\frac{1576}{133}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &-\frac{44}{71}\ &-\frac{68}{213}\ &-\frac{277}{213}\ &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.