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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\ &3\ &3\ &3\ &3\ &3\ \\ &3\ &4\ &3\ &3\ &3\ &3\ \\ &3\ &3\ &4\ &3\ &3\ &3\ \\ &3\ &3\ &3\ &4\ &3\ &3\ \\ &3\ &3\ &3\ &3\ &4\ &3\ \\ &3\ &3\ &3\ &3\ &3\ &4\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &4\ &3\ &3\ &3\ &3\ &3\ \\ &3\ &4\ &3\ &3\ &3\ &3\ \\ &3\ &3\ &4\ &3\ &3\ &3\ \\ &3\ &3\ &3\ &4\ &3\ &3\ \\ &3\ &3\ &3\ &3\ &4\ &3\ \\ &3\ &3\ &3\ &3\ &3\ &4\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &3\ &3\ &3\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &3\ &4\ &3\ &3\ &3\ &3\ &0\ &1\ &0\ &0\ &0\ &0\ \\ &3\ &3\ &4\ &3\ &3\ &3\ &0\ &0\ &1\ &0\ &0\ &0\ \\ &3\ &3\ &3\ &4\ &3\ &3\ &0\ &0\ &0\ &1\ &0\ &0\ \\ &3\ &3\ &3\ &3\ &4\ &3\ &0\ &0\ &0\ &0\ &1\ &0\ \\ &3\ &3\ &3\ &3\ &3\ &4\ &0\ &0\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &3\ &3\ &3\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &-\frac{3}{4}\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{3}{4}\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &-\frac{3}{4}\ &0\ &1\ &0\ &0\ &0\ \\ &0\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &-\frac{3}{4}\ &0\ &0\ &1\ &0\ &0\ \\ &0\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{7}{4}\ &\frac{3}{4}\ &-\frac{3}{4}\ &0\ &0\ &0\ &1\ &0\ \\ &0\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{7}{4}\ &-\frac{3}{4}\ &0\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &3\ &3\ &3\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &-\frac{3}{4}\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{10}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &-\frac{3}{7}\ &-\frac{3}{7}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{3}{7}\ &\frac{10}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &-\frac{3}{7}\ &-\frac{3}{7}\ &0\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{3}{7}\ &\frac{3}{7}\ &\frac{10}{7}\ &\frac{3}{7}\ &-\frac{3}{7}\ &-\frac{3}{7}\ &0\ &0\ &1\ &0\ \\ &0\ &0\ &\frac{3}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &\frac{10}{7}\ &-\frac{3}{7}\ &-\frac{3}{7}\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &3\ &3\ &3\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &-\frac{3}{4}\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{10}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &-\frac{3}{7}\ &-\frac{3}{7}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &\frac{3}{10}\ &\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{3}{10}\ &\frac{13}{10}\ &\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &0\ &1\ &0\ \\ &0\ &0\ &0\ &\frac{3}{10}\ &\frac{3}{10}\ &\frac{13}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &3\ &3\ &3\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &-\frac{3}{4}\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{10}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &-\frac{3}{7}\ &-\frac{3}{7}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &\frac{3}{10}\ &\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &\frac{3}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &\frac{3}{13}\ &\frac{16}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &3\ &3\ &3\ &1\ &0\ &0\ &0\ &0\ &0\ \\ &0\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &-\frac{3}{4}\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{10}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &-\frac{3}{7}\ &-\frac{3}{7}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &\frac{3}{10}\ &\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &-\frac{3}{10}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &\frac{3}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &-\frac{3}{13}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ above\ the\ diagonal\ to\ 0}\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &3\ &3\ &0\ &\frac{28}{19}\ &\frac{9}{19}\ &\frac{9}{19}\ &\frac{9}{19}\ &\frac{9}{19}\ &-\frac{48}{19}\ \\ &0\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &0\ &-\frac{12}{19}\ &\frac{85}{76}\ &\frac{9}{76}\ &\frac{9}{76}\ &\frac{9}{76}\ &-\frac{12}{19}\ \\ &0\ &0\ &\frac{10}{7}\ &\frac{3}{7}\ &\frac{3}{7}\ &0\ &-\frac{48}{133}\ &-\frac{48}{133}\ &\frac{142}{133}\ &\frac{9}{133}\ &\frac{9}{133}\ &-\frac{48}{133}\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &\frac{3}{10}\ &0\ &-\frac{24}{95}\ &-\frac{24}{95}\ &-\frac{24}{95}\ &\frac{199}{190}\ &\frac{9}{190}\ &-\frac{24}{95}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &3\ &0\ &0\ &\frac{37}{19}\ &\frac{18}{19}\ &\frac{18}{19}\ &\frac{18}{19}\ &-\frac{39}{19}\ &-\frac{39}{19}\ \\ &0\ &\frac{7}{4}\ &\frac{3}{4}\ &\frac{3}{4}\ &0\ &0\ &-\frac{39}{76}\ &\frac{47}{38}\ &\frac{9}{38}\ &\frac{9}{38}\ &-\frac{39}{76}\ &-\frac{39}{76}\ \\ &0\ &0\ &\frac{10}{7}\ &\frac{3}{7}\ &0\ &0\ &-\frac{39}{133}\ &-\frac{39}{133}\ &\frac{151}{133}\ &\frac{18}{133}\ &-\frac{39}{133}\ &-\frac{39}{133}\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &0\ &0\ &-\frac{39}{190}\ &-\frac{39}{190}\ &-\frac{39}{190}\ &\frac{104}{95}\ &-\frac{39}{190}\ &-\frac{39}{190}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &3\ &0\ &0\ &0\ &\frac{46}{19}\ &\frac{27}{19}\ &\frac{27}{19}\ &-\frac{30}{19}\ &-\frac{30}{19}\ &-\frac{30}{19}\ \\ &0\ &\frac{7}{4}\ &\frac{3}{4}\ &0\ &0\ &0\ &-\frac{15}{38}\ &\frac{103}{76}\ &\frac{27}{76}\ &-\frac{15}{38}\ &-\frac{15}{38}\ &-\frac{15}{38}\ \\ &0\ &0\ &\frac{10}{7}\ &0\ &0\ &0\ &-\frac{30}{133}\ &-\frac{30}{133}\ &\frac{160}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &0\ &0\ &-\frac{39}{190}\ &-\frac{39}{190}\ &-\frac{39}{190}\ &\frac{104}{95}\ &-\frac{39}{190}\ &-\frac{39}{190}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &3\ &0\ &0\ &0\ &0\ &\frac{55}{19}\ &\frac{36}{19}\ &-\frac{21}{19}\ &-\frac{21}{19}\ &-\frac{21}{19}\ &-\frac{21}{19}\ \\ &0\ &\frac{7}{4}\ &0\ &0\ &0\ &0\ &-\frac{21}{76}\ &\frac{28}{19}\ &-\frac{21}{76}\ &-\frac{21}{76}\ &-\frac{21}{76}\ &-\frac{21}{76}\ \\ &0\ &0\ &\frac{10}{7}\ &0\ &0\ &0\ &-\frac{30}{133}\ &-\frac{30}{133}\ &\frac{160}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &0\ &0\ &-\frac{39}{190}\ &-\frac{39}{190}\ &-\frac{39}{190}\ &\frac{104}{95}\ &-\frac{39}{190}\ &-\frac{39}{190}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &4\ &0\ &0\ &0\ &0\ &0\ &\frac{64}{19}\ &-\frac{12}{19}\ &-\frac{12}{19}\ &-\frac{12}{19}\ &-\frac{12}{19}\ &-\frac{12}{19}\ \\ &0\ &\frac{7}{4}\ &0\ &0\ &0\ &0\ &-\frac{21}{76}\ &\frac{28}{19}\ &-\frac{21}{76}\ &-\frac{21}{76}\ &-\frac{21}{76}\ &-\frac{21}{76}\ \\ &0\ &0\ &\frac{10}{7}\ &0\ &0\ &0\ &-\frac{30}{133}\ &-\frac{30}{133}\ &\frac{160}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &0\ &0\ &-\frac{39}{190}\ &-\frac{39}{190}\ &-\frac{39}{190}\ &\frac{104}{95}\ &-\frac{39}{190}\ &-\frac{39}{190}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ on\ the\ main\ diagonal\ to\ 1}\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &1\ &0\ &0\ &0\ &0\ &0\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &\frac{7}{4}\ &0\ &0\ &0\ &0\ &-\frac{21}{76}\ &\frac{28}{19}\ &-\frac{21}{76}\ &-\frac{21}{76}\ &-\frac{21}{76}\ &-\frac{21}{76}\ \\ &0\ &0\ &\frac{10}{7}\ &0\ &0\ &0\ &-\frac{30}{133}\ &-\frac{30}{133}\ &\frac{160}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &0\ &0\ &-\frac{39}{190}\ &-\frac{39}{190}\ &-\frac{39}{190}\ &\frac{104}{95}\ &-\frac{39}{190}\ &-\frac{39}{190}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &1\ &0\ &0\ &0\ &0\ &0\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &1\ &0\ &0\ &0\ &0\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &\frac{10}{7}\ &0\ &0\ &0\ &-\frac{30}{133}\ &-\frac{30}{133}\ &\frac{160}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ &-\frac{30}{133}\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &0\ &0\ &-\frac{39}{190}\ &-\frac{39}{190}\ &-\frac{39}{190}\ &\frac{104}{95}\ &-\frac{39}{190}\ &-\frac{39}{190}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &1\ &0\ &0\ &0\ &0\ &0\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &1\ &0\ &0\ &0\ &0\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &1\ &0\ &0\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &\frac{13}{10}\ &0\ &0\ &-\frac{39}{190}\ &-\frac{39}{190}\ &-\frac{39}{190}\ &\frac{104}{95}\ &-\frac{39}{190}\ &-\frac{39}{190}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &1\ &0\ &0\ &0\ &0\ &0\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &1\ &0\ &0\ &0\ &0\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &1\ &0\ &0\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &1\ &0\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &0\ &\frac{16}{13}\ &0\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &-\frac{48}{247}\ &\frac{256}{247}\ &-\frac{48}{247}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &1\ &0\ &0\ &0\ &0\ &0\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &1\ &0\ &0\ &0\ &0\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &1\ &0\ &0\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &1\ &0\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &0\ &1\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &0\ &0\ &\frac{19}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &-\frac{3}{16}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccccc | cccccc} &1\ &0\ &0\ &0\ &0\ &0\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &1\ &0\ &0\ &0\ &0\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &1\ &0\ &0\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &1\ &0\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &0\ &1\ &0\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ \\ &0\ &0\ &0\ &0\ &0\ &1\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ \\\end{array} \right )\\\\&\color{grey}{The\ inverse\ matrix\ obtained\ is\ : }\\&\begin{pmatrix} &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ \\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ &-\frac{3}{19}\ \\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &-\frac{3}{19}\ &\frac{16}{19}\ \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.