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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} &5\ &6\ &8\ &10\ &12\ \\ &6\ &8\ &10\ &12\ &15\ \\ &8\ &12\ &15\ &20\ &24\ \\ &12\ &15\ &20\ &24\ &25\ \\ &15\ &20\ &24\ &25\ &30\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &5\ &6\ &8\ &10\ &12\ \\ &6\ &8\ &10\ &12\ &15\ \\ &8\ &12\ &15\ &20\ &24\ \\ &12\ &15\ &20\ &24\ &25\ \\ &15\ &20\ &24\ &25\ &30\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {cccccc | ccccc} &5\ &6\ &8\ &10\ &12\ &1\ &0\ &0\ &0\ &0\ \\ &6\ &8\ &10\ &12\ &15\ &0\ &1\ &0\ &0\ &0\ \\ &8\ &12\ &15\ &20\ &24\ &0\ &0\ &1\ &0\ &0\ \\ &12\ &15\ &20\ &24\ &25\ &0\ &0\ &0\ &1\ &0\ \\ &15\ &20\ &24\ &25\ &30\ &0\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &5\ &6\ &8\ &10\ &12\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{4}{5}\ &\frac{2}{5}\ &0\ &\frac{3}{5}\ &-\frac{6}{5}\ &1\ &0\ &0\ &0\ \\ &0\ &\frac{12}{5}\ &\frac{11}{5}\ &4\ &\frac{24}{5}\ &-\frac{8}{5}\ &0\ &1\ &0\ &0\ \\ &0\ &\frac{3}{5}\ &\frac{4}{5}\ &0\ &-\frac{19}{5}\ &-\frac{12}{5}\ &0\ &0\ &1\ &0\ \\ &0\ &2\ &0\ &-5\ &-6\ &-3\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &5\ &6\ &8\ &10\ &12\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{4}{5}\ &\frac{2}{5}\ &0\ &\frac{3}{5}\ &-\frac{6}{5}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &1\ &4\ &3\ &2\ &-3\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{1}{2}\ &0\ &-\frac{17}{4}\ &-\frac{3}{2}\ &-\frac{3}{4}\ &0\ &1\ &0\ \\ &0\ &0\ &-1\ &-5\ &-\frac{15}{2}\ &0\ &-\frac{5}{2}\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &5\ &6\ &8\ &10\ &12\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{4}{5}\ &\frac{2}{5}\ &0\ &\frac{3}{5}\ &-\frac{6}{5}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &1\ &4\ &3\ &2\ &-3\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &-2\ &-\frac{23}{4}\ &-\frac{5}{2}\ &\frac{3}{4}\ &-\frac{1}{2}\ &1\ &0\ \\ &0\ &0\ &0\ &-1\ &-\frac{9}{2}\ &2\ &-\frac{11}{2}\ &1\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &5\ &6\ &8\ &10\ &12\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{4}{5}\ &\frac{2}{5}\ &0\ &\frac{3}{5}\ &-\frac{6}{5}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &1\ &4\ &3\ &2\ &-3\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &-2\ &-\frac{23}{4}\ &-\frac{5}{2}\ &\frac{3}{4}\ &-\frac{1}{2}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &-\frac{13}{8}\ &\frac{13}{4}\ &-\frac{47}{8}\ &\frac{5}{4}\ &-\frac{1}{2}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ above\ the\ diagonal\ to\ 0}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &5\ &6\ &8\ &10\ &0\ &25\ &-\frac{564}{13}\ &\frac{120}{13}\ &-\frac{48}{13}\ &\frac{96}{13}\ \\ &0\ &\frac{4}{5}\ &\frac{2}{5}\ &0\ &0\ &0\ &-\frac{76}{65}\ &\frac{6}{13}\ &-\frac{12}{65}\ &\frac{24}{65}\ \\ &0\ &0\ &1\ &4\ &0\ &8\ &-\frac{180}{13}\ &\frac{43}{13}\ &-\frac{12}{13}\ &\frac{24}{13}\ \\ &0\ &0\ &0\ &-2\ &0\ &-14\ &\frac{280}{13}\ &-\frac{64}{13}\ &\frac{36}{13}\ &-\frac{46}{13}\ \\ &0\ &0\ &0\ &0\ &-\frac{13}{8}\ &\frac{13}{4}\ &-\frac{47}{8}\ &\frac{5}{4}\ &-\frac{1}{2}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &5\ &6\ &8\ &0\ &0\ &-45\ &\frac{836}{13}\ &-\frac{200}{13}\ &\frac{132}{13}\ &-\frac{134}{13}\ \\ &0\ &\frac{4}{5}\ &\frac{2}{5}\ &0\ &0\ &0\ &-\frac{76}{65}\ &\frac{6}{13}\ &-\frac{12}{65}\ &\frac{24}{65}\ \\ &0\ &0\ &1\ &0\ &0\ &-20\ &\frac{380}{13}\ &-\frac{85}{13}\ &\frac{60}{13}\ &-\frac{68}{13}\ \\ &0\ &0\ &0\ &-2\ &0\ &-14\ &\frac{280}{13}\ &-\frac{64}{13}\ &\frac{36}{13}\ &-\frac{46}{13}\ \\ &0\ &0\ &0\ &0\ &-\frac{13}{8}\ &\frac{13}{4}\ &-\frac{47}{8}\ &\frac{5}{4}\ &-\frac{1}{2}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &5\ &6\ &0\ &0\ &0\ &115\ &-\frac{2204}{13}\ &\frac{480}{13}\ &-\frac{348}{13}\ &\frac{410}{13}\ \\ &0\ &\frac{4}{5}\ &0\ &0\ &0\ &8\ &-\frac{836}{65}\ &\frac{40}{13}\ &-\frac{132}{65}\ &\frac{32}{13}\ \\ &0\ &0\ &1\ &0\ &0\ &-20\ &\frac{380}{13}\ &-\frac{85}{13}\ &\frac{60}{13}\ &-\frac{68}{13}\ \\ &0\ &0\ &0\ &-2\ &0\ &-14\ &\frac{280}{13}\ &-\frac{64}{13}\ &\frac{36}{13}\ &-\frac{46}{13}\ \\ &0\ &0\ &0\ &0\ &-\frac{13}{8}\ &\frac{13}{4}\ &-\frac{47}{8}\ &\frac{5}{4}\ &-\frac{1}{2}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &5\ &0\ &0\ &0\ &0\ &55\ &-\frac{950}{13}\ &\frac{180}{13}\ &-\frac{150}{13}\ &\frac{170}{13}\ \\ &0\ &\frac{4}{5}\ &0\ &0\ &0\ &8\ &-\frac{836}{65}\ &\frac{40}{13}\ &-\frac{132}{65}\ &\frac{32}{13}\ \\ &0\ &0\ &1\ &0\ &0\ &-20\ &\frac{380}{13}\ &-\frac{85}{13}\ &\frac{60}{13}\ &-\frac{68}{13}\ \\ &0\ &0\ &0\ &-2\ &0\ &-14\ &\frac{280}{13}\ &-\frac{64}{13}\ &\frac{36}{13}\ &-\frac{46}{13}\ \\ &0\ &0\ &0\ &0\ &-\frac{13}{8}\ &\frac{13}{4}\ &-\frac{47}{8}\ &\frac{5}{4}\ &-\frac{1}{2}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ on\ the\ main\ diagonal\ to\ 1}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &11\ &-\frac{190}{13}\ &\frac{36}{13}\ &-\frac{30}{13}\ &\frac{34}{13}\ \\ &0\ &\frac{4}{5}\ &0\ &0\ &0\ &8\ &-\frac{836}{65}\ &\frac{40}{13}\ &-\frac{132}{65}\ &\frac{32}{13}\ \\ &0\ &0\ &1\ &0\ &0\ &-20\ &\frac{380}{13}\ &-\frac{85}{13}\ &\frac{60}{13}\ &-\frac{68}{13}\ \\ &0\ &0\ &0\ &-2\ &0\ &-14\ &\frac{280}{13}\ &-\frac{64}{13}\ &\frac{36}{13}\ &-\frac{46}{13}\ \\ &0\ &0\ &0\ &0\ &-\frac{13}{8}\ &\frac{13}{4}\ &-\frac{47}{8}\ &\frac{5}{4}\ &-\frac{1}{2}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &11\ &-\frac{190}{13}\ &\frac{36}{13}\ &-\frac{30}{13}\ &\frac{34}{13}\ \\ &0\ &1\ &0\ &0\ &0\ &10\ &-\frac{209}{13}\ &\frac{50}{13}\ &-\frac{33}{13}\ &\frac{40}{13}\ \\ &0\ &0\ &1\ &0\ &0\ &-20\ &\frac{380}{13}\ &-\frac{85}{13}\ &\frac{60}{13}\ &-\frac{68}{13}\ \\ &0\ &0\ &0\ &-2\ &0\ &-14\ &\frac{280}{13}\ &-\frac{64}{13}\ &\frac{36}{13}\ &-\frac{46}{13}\ \\ &0\ &0\ &0\ &0\ &-\frac{13}{8}\ &\frac{13}{4}\ &-\frac{47}{8}\ &\frac{5}{4}\ &-\frac{1}{2}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &11\ &-\frac{190}{13}\ &\frac{36}{13}\ &-\frac{30}{13}\ &\frac{34}{13}\ \\ &0\ &1\ &0\ &0\ &0\ &10\ &-\frac{209}{13}\ &\frac{50}{13}\ &-\frac{33}{13}\ &\frac{40}{13}\ \\ &0\ &0\ &1\ &0\ &0\ &-20\ &\frac{380}{13}\ &-\frac{85}{13}\ &\frac{60}{13}\ &-\frac{68}{13}\ \\ &0\ &0\ &0\ &1\ &0\ &7\ &-\frac{140}{13}\ &\frac{32}{13}\ &-\frac{18}{13}\ &\frac{23}{13}\ \\ &0\ &0\ &0\ &0\ &-\frac{13}{8}\ &\frac{13}{4}\ &-\frac{47}{8}\ &\frac{5}{4}\ &-\frac{1}{2}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &11\ &-\frac{190}{13}\ &\frac{36}{13}\ &-\frac{30}{13}\ &\frac{34}{13}\ \\ &0\ &1\ &0\ &0\ &0\ &10\ &-\frac{209}{13}\ &\frac{50}{13}\ &-\frac{33}{13}\ &\frac{40}{13}\ \\ &0\ &0\ &1\ &0\ &0\ &-20\ &\frac{380}{13}\ &-\frac{85}{13}\ &\frac{60}{13}\ &-\frac{68}{13}\ \\ &0\ &0\ &0\ &1\ &0\ &7\ &-\frac{140}{13}\ &\frac{32}{13}\ &-\frac{18}{13}\ &\frac{23}{13}\ \\ &0\ &0\ &0\ &0\ &1\ &-2\ &\frac{47}{13}\ &-\frac{10}{13}\ &\frac{4}{13}\ &-\frac{8}{13}\ \\\end{array} \right )\\\\&\color{grey}{The\ inverse\ matrix\ obtained\ is\ : }\\&\begin{pmatrix} &11\ &-\frac{190}{13}\ &\frac{36}{13}\ &-\frac{30}{13}\ &\frac{34}{13}\ \\ &10\ &-\frac{209}{13}\ &\frac{50}{13}\ &-\frac{33}{13}\ &\frac{40}{13}\ \\ &-20\ &\frac{380}{13}\ &-\frac{85}{13}\ &\frac{60}{13}\ &-\frac{68}{13}\ \\ &7\ &-\frac{140}{13}\ &\frac{32}{13}\ &-\frac{18}{13}\ &\frac{23}{13}\ \\ &-2\ &\frac{47}{13}\ &-\frac{10}{13}\ &\frac{4}{13}\ &-\frac{8}{13}\ \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.