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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} &4\ &2\ &4\ &2\ \\ &8\ &16\ &8\ &4\ \\ &32\ &64\ &128\ &16\ \\ &256\ &512\ &2048\ &4096\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &4\ &2\ &4\ &2\ \\ &8\ &16\ &8\ &4\ \\ &32\ &64\ &128\ &16\ \\ &256\ &512\ &2048\ &4096\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {ccccc | cccc} &4\ &2\ &4\ &2\ &1\ &0\ &0\ &0\ \\ &8\ &16\ &8\ &4\ &0\ &1\ &0\ &0\ \\ &32\ &64\ &128\ &16\ &0\ &0\ &1\ &0\ \\ &256\ &512\ &2048\ &4096\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &4\ &2\ &4\ &2\ &1\ &0\ &0\ &0\ \\ &0\ &12\ &0\ &0\ &-2\ &1\ &0\ &0\ \\ &0\ &48\ &96\ &0\ &-8\ &0\ &1\ &0\ \\ &0\ &384\ &1792\ &3968\ &-64\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &4\ &2\ &4\ &2\ &1\ &0\ &0\ &0\ \\ &0\ &12\ &0\ &0\ &-2\ &1\ &0\ &0\ \\ &0\ &0\ &96\ &0\ &0\ &-4\ &1\ &0\ \\ &0\ &0\ &1792\ &3968\ &0\ &-32\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &4\ &2\ &4\ &2\ &1\ &0\ &0\ &0\ \\ &0\ &12\ &0\ &0\ &-2\ &1\ &0\ &0\ \\ &0\ &0\ &96\ &0\ &0\ &-4\ &1\ &0\ \\ &0\ &0\ &0\ &3968\ &0\ &\frac{128}{3}\ &-\frac{56}{3}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ above\ the\ diagonal\ to\ 0}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &4\ &2\ &4\ &0\ &1\ &-\frac{2}{93}\ &\frac{7}{744}\ &-\frac{1}{1984}\ \\ &0\ &12\ &0\ &0\ &-2\ &1\ &0\ &0\ \\ &0\ &0\ &96\ &0\ &0\ &-4\ &1\ &0\ \\ &0\ &0\ &0\ &3968\ &0\ &\frac{128}{3}\ &-\frac{56}{3}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &4\ &2\ &0\ &0\ &1\ &\frac{9}{62}\ &-\frac{1}{31}\ &-\frac{1}{1984}\ \\ &0\ &12\ &0\ &0\ &-2\ &1\ &0\ &0\ \\ &0\ &0\ &96\ &0\ &0\ &-4\ &1\ &0\ \\ &0\ &0\ &0\ &3968\ &0\ &\frac{128}{3}\ &-\frac{56}{3}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &4\ &0\ &0\ &0\ &\frac{4}{3}\ &-\frac{2}{93}\ &-\frac{1}{31}\ &-\frac{1}{1984}\ \\ &0\ &12\ &0\ &0\ &-2\ &1\ &0\ &0\ \\ &0\ &0\ &96\ &0\ &0\ &-4\ &1\ &0\ \\ &0\ &0\ &0\ &3968\ &0\ &\frac{128}{3}\ &-\frac{56}{3}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ on\ the\ main\ diagonal\ to\ 1}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &\frac{1}{3}\ &-\frac{1}{186}\ &-\frac{1}{124}\ &-\frac{1}{7936}\ \\ &0\ &12\ &0\ &0\ &-2\ &1\ &0\ &0\ \\ &0\ &0\ &96\ &0\ &0\ &-4\ &1\ &0\ \\ &0\ &0\ &0\ &3968\ &0\ &\frac{128}{3}\ &-\frac{56}{3}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &\frac{1}{3}\ &-\frac{1}{186}\ &-\frac{1}{124}\ &-\frac{1}{7936}\ \\ &0\ &1\ &0\ &0\ &-\frac{1}{6}\ &\frac{1}{12}\ &0\ &0\ \\ &0\ &0\ &96\ &0\ &0\ &-4\ &1\ &0\ \\ &0\ &0\ &0\ &3968\ &0\ &\frac{128}{3}\ &-\frac{56}{3}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &\frac{1}{3}\ &-\frac{1}{186}\ &-\frac{1}{124}\ &-\frac{1}{7936}\ \\ &0\ &1\ &0\ &0\ &-\frac{1}{6}\ &\frac{1}{12}\ &0\ &0\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{1}{24}\ &\frac{1}{96}\ &0\ \\ &0\ &0\ &0\ &3968\ &0\ &\frac{128}{3}\ &-\frac{56}{3}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &\frac{1}{3}\ &-\frac{1}{186}\ &-\frac{1}{124}\ &-\frac{1}{7936}\ \\ &0\ &1\ &0\ &0\ &-\frac{1}{6}\ &\frac{1}{12}\ &0\ &0\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{1}{24}\ &\frac{1}{96}\ &0\ \\ &0\ &0\ &0\ &1\ &0\ &\frac{1}{93}\ &-\frac{7}{1488}\ &\frac{1}{3968}\ \\\end{array} \right )\\\\&\color{grey}{The\ inverse\ matrix\ obtained\ is\ : }\\&\begin{pmatrix} &\frac{1}{3}\ &-\frac{1}{186}\ &-\frac{1}{124}\ &-\frac{1}{7936}\ \\ &-\frac{1}{6}\ &\frac{1}{12}\ &0\ &0\ \\ &0\ &-\frac{1}{24}\ &\frac{1}{96}\ &0\ \\ &0\ &\frac{1}{93}\ &-\frac{7}{1488}\ &\frac{1}{3968}\ \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.