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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} &1\ &43\ &26\ &32\ \\ &74\ &37\ &59\ &86\ \\ &24\ &56\ &85\ &37\ \\ &96\ &45\ &28\ &43\ \end{pmatrix}\color{black}{\ .}\\ \\Solu&tion:\\ &\begin{pmatrix} &1\ &43\ &26\ &32\ \\ &74\ &37\ &59\ &86\ \\ &24\ &56\ &85\ &37\ \\ &96\ &45\ &28\ &43\ \end{pmatrix}\\\\&\color{grey}{Using\ the\ elementary\ transformation\ of\ the\ matrix\ to\ find\ the\ inverse\ matrix:}\\&\left (\begin{array} {ccccc | cccc} &1\ &43\ &26\ &32\ &1\ &0\ &0\ &0\ \\ &74\ &37\ &59\ &86\ &0\ &1\ &0\ &0\ \\ &24\ &56\ &85\ &37\ &0\ &0\ &1\ &0\ \\ &96\ &45\ &28\ &43\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{Transfprming\ a\ known\ matrix\ into\ an\ upper\ triangular\ matrix :}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &43\ &26\ &32\ &1\ &0\ &0\ &0\ \\ &0\ &-3145\ &-1865\ &-2282\ &-74\ &1\ &0\ &0\ \\ &0\ &-976\ &-539\ &-731\ &-24\ &0\ &1\ &0\ \\ &0\ &-4083\ &-2468\ &-3029\ &-96\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &43\ &26\ &32\ &1\ &0\ &0\ &0\ \\ &0\ &-3145\ &-1865\ &-2282\ &-74\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{25017}{629}\ &-\frac{71763}{3145}\ &-\frac{88}{85}\ &-\frac{976}{3145}\ &1\ &0\ \\ &0\ &0\ &-\frac{29413}{629}\ &-\frac{208799}{3145}\ &\frac{6}{85}\ &-\frac{4083}{3145}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &43\ &26\ &32\ &1\ &0\ &0\ &0\ \\ &0\ &-3145\ &-1865\ &-2282\ &-74\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{25017}{629}\ &-\frac{71763}{3145}\ &-\frac{88}{85}\ &-\frac{976}{3145}\ &1\ &0\ \\ &0\ &0\ &0\ &-\frac{3886746}{41695}\ &-\frac{143426}{125085}\ &-\frac{208031}{125085}\ &\frac{29413}{25017}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ above\ the\ diagonal\ to\ 0}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &43\ &26\ &0\ &\frac{3535303}{5830119}\ &-\frac{3328496}{5830119}\ &\frac{2353040}{5830119}\ &\frac{667120}{1943373}\ \\ &0\ &-3145\ &-1865\ &0\ &-\frac{267779740}{5830119}\ &\frac{243193490}{5830119}\ &-\frac{167801165}{5830119}\ &-\frac{47573995}{1943373}\ \\ &0\ &0\ &\frac{25017}{629}\ &0\ &-\frac{922435163}{1222381617}\ &\frac{236569091}{2444763234}\ &\frac{1741174861}{2444763234}\ &-\frac{199477219}{814921078}\ \\ &0\ &0\ &0\ &-\frac{3886746}{41695}\ &-\frac{143426}{125085}\ &-\frac{208031}{125085}\ &\frac{29413}{25017}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &43\ &0\ &0\ &\frac{2137115}{1943373}\ &-\frac{1232431}{1943373}\ &-\frac{120449}{1943373}\ &\frac{326031}{647791}\ \\ &0\ &-3145\ &0\ &0\ &-\frac{52675605}{647791}\ &\frac{59921685}{1295582}\ &\frac{5978645}{1295582}\ &-\frac{46586885}{1295582}\ \\ &0\ &0\ &\frac{25017}{629}\ &0\ &-\frac{922435163}{1222381617}\ &\frac{236569091}{2444763234}\ &\frac{1741174861}{2444763234}\ &-\frac{199477219}{814921078}\ \\ &0\ &0\ &0\ &-\frac{3886746}{41695}\ &-\frac{143426}{125085}\ &-\frac{208031}{125085}\ &\frac{29413}{25017}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &-\frac{23506}{1943373}\ &-\frac{7025}{3886746}\ &\frac{4331}{3886746}\ &\frac{15103}{1295582}\ \\ &0\ &-3145\ &0\ &0\ &-\frac{52675605}{647791}\ &\frac{59921685}{1295582}\ &\frac{5978645}{1295582}\ &-\frac{46586885}{1295582}\ \\ &0\ &0\ &\frac{25017}{629}\ &0\ &-\frac{922435163}{1222381617}\ &\frac{236569091}{2444763234}\ &\frac{1741174861}{2444763234}\ &-\frac{199477219}{814921078}\ \\ &0\ &0\ &0\ &-\frac{3886746}{41695}\ &-\frac{143426}{125085}\ &-\frac{208031}{125085}\ &\frac{29413}{25017}\ &1\ \\\end{array} \right )\\\\&\color{grey}{Convert\ elements\ on\ the\ main\ diagonal\ to\ 1}\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &-\frac{23506}{1943373}\ &-\frac{7025}{3886746}\ &\frac{4331}{3886746}\ &\frac{15103}{1295582}\ \\ &0\ &1\ &0\ &0\ &\frac{16749}{647791}\ &-\frac{19053}{1295582}\ &-\frac{1901}{1295582}\ &\frac{14813}{1295582}\ \\ &0\ &0\ &\frac{25017}{629}\ &0\ &-\frac{922435163}{1222381617}\ &\frac{236569091}{2444763234}\ &\frac{1741174861}{2444763234}\ &-\frac{199477219}{814921078}\ \\ &0\ &0\ &0\ &-\frac{3886746}{41695}\ &-\frac{143426}{125085}\ &-\frac{208031}{125085}\ &\frac{29413}{25017}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &-\frac{23506}{1943373}\ &-\frac{7025}{3886746}\ &\frac{4331}{3886746}\ &\frac{15103}{1295582}\ \\ &0\ &1\ &0\ &0\ &\frac{16749}{647791}\ &-\frac{19053}{1295582}\ &-\frac{1901}{1295582}\ &\frac{14813}{1295582}\ \\ &0\ &0\ &1\ &0\ &-\frac{110617}{5830119}\ &\frac{28369}{11660238}\ &\frac{208799}{11660238}\ &-\frac{23921}{3886746}\ \\ &0\ &0\ &0\ &-\frac{3886746}{41695}\ &-\frac{143426}{125085}\ &-\frac{208031}{125085}\ &\frac{29413}{25017}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {ccccc | cccc} &1\ &0\ &0\ &0\ &-\frac{23506}{1943373}\ &-\frac{7025}{3886746}\ &\frac{4331}{3886746}\ &\frac{15103}{1295582}\ \\ &0\ &1\ &0\ &0\ &\frac{16749}{647791}\ &-\frac{19053}{1295582}\ &-\frac{1901}{1295582}\ &\frac{14813}{1295582}\ \\ &0\ &0\ &1\ &0\ &-\frac{110617}{5830119}\ &\frac{28369}{11660238}\ &\frac{208799}{11660238}\ &-\frac{23921}{3886746}\ \\ &0\ &0\ &0\ &\frac{1163}{1163}\ &\frac{71713}{5830119}\ &\frac{208031}{11660238}\ &-\frac{147065}{11660238}\ &-\frac{41695}{3886746}\ \\\end{array} \right )\\\\&\color{grey}{The\ inverse\ matrix\ obtained\ is\ : }\\&\begin{pmatrix} &-\frac{23506}{1943373}\ &-\frac{7025}{3886746}\ &\frac{4331}{3886746}\ &\frac{15103}{1295582}\ \\ &\frac{16749}{647791}\ &-\frac{19053}{1295582}\ &-\frac{1901}{1295582}\ &\frac{14813}{1295582}\ \\ &-\frac{110617}{5830119}\ &\frac{28369}{11660238}\ &\frac{208799}{11660238}\ &-\frac{23921}{3886746}\ \\ &\frac{71713}{5830119}\ &\frac{208031}{11660238}\ &-\frac{147065}{11660238}\ &-\frac{41695}{3886746}\ \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.