Mathematics-I (3110014)

$\mathrm{Here}, \mathrm{A} = \left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 3\\ 1& 0& 8\end{array}\right] .$

$\mathrm{Now}, \left|\mathrm{A}\right| = \left|\begin{array}{ccc}1& 2& 3\\ 2& 5& 3\\ 1& 0& 8\end{array}\right|$

$\mathrm{Now}, \left|\mathrm{A}\right| = 1(40-0)-2(16-3)+3(0-5)$

$\mathrm{Now}, \left|\mathrm{A}\right| = 40-26-15$

$\mathrm{Now}, \left|\mathrm{A}\right| = -1\ne 0$

$\mathrm{So}, {\mathrm{A}}^{-1} \mathrm{exist}.$

$\mathrm{By} \mathrm{Gauss} - \mathrm{Jordan} \mathrm{Method},$

$\left[\overline{)\mathrm{A} } \mathrm{I} \right]=\left[\overline{)\begin{array}{ccc}1& 2& 3\\ 2& 5& 3\\ 1& 0& 8\end{array} } \begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array} \right]$

$\mathrm{Apply},{\mathrm{R}}_{12} (-2) , {\mathrm{R}}_{13} (-1)$

$\left[\overline{)\mathrm{A} } \mathrm{I} \right] ~ \left[\overline{)\begin{array}{ccc}1& 2& 3\\ 2-2\left(1\right)& 5-2\left(2\right)& 3-2\left(3\right)\\ 1-1\left(1\right)& 0-1\left(2\right)& 8-1\left(3\right)\end{array} } \begin{array}{ccc}1& 0& 0\\ 0-2\left(1\right)& 1-2\left(0\right)& 0-2\left(0\right)\\ 0-1\left(1\right)& 0-1\left(0\right)& 1-1\left(0\right)\end{array} \right]$

$\left[\overline{)\mathrm{A} } \mathrm{I} \right] ~ \left[\overline{)\begin{array}{ccc}1& 2& 3\\ 0& 1& -3\\ 0& -2& 5\end{array} } \begin{array}{ccc} 1& 0& 0\\ -2& 1 & 0 \\ -1& 0& 1\end{array} \right]$

$\mathrm{Apply},{\mathrm{R}}_{23} \left(2\right) , {\mathrm{R}}_{21} (-2)$

$\left[\overline{)\mathrm{A} } \mathrm{I} \right] ~ \left[\overline{)\begin{array}{ccc}1-2\left(0\right)& 2-2\left(1\right)& 3-2(-3)\\ 0& 1& -3\\ 0+2\left(0\right)& -2+2\left(1\right)& 5+2(-3)\end{array} } \begin{array}{ccc}1-2(-2)& 0-2\left(1\right)& 0-2\left(0\right)\\ -2& 1 & 0 \\ -1+2(-2)& 0+2\left(1\right)& 1+2\left(0\right)\end{array} \right]$

$\left[\overline{)\mathrm{A} } \mathrm{I} \right] ~ \left[\overline{)\begin{array}{ccc}1& 0& 9\\ 0& 1& -3 \\ 0& 0& -1\end{array} } \begin{array}{ccc} 5& -2& 0\\ -2& 1 & 0\\ -5& 2& 1\end{array} \right]$

$\mathrm{Apply}, {\mathrm{R}}_3(-1)$

$\left[\overline{)\mathrm{A} } \mathrm{I} \right] ~ \left[\overline{)\begin{array}{ccc}1& 0& 9\\ 0& 1& -3 \\ 0& 0& 1\end{array} } \begin{array}{ccc} 5& -2& 0\\ -2& 1 & 0\\ 5& -2& -1\end{array} \right]$

$\mathrm{Apply},{\mathrm{R}}_{32} \left(3\right) , {\mathrm{R}}_{31} (-9)$

$\left[\overline{)\mathrm{A} } \mathrm{I} \right] ~ \left[\overline{)\begin{array}{ccc}1-9\left(0\right)& 0-9\left(0\right)& 9-9\left(1\right)\\ 0+3\left(0\right)& 1+3\left(0\right)& -3+3\left(1\right)\\ 0& 0& 1\end{array} } \begin{array}{ccc}5-9\left(5\right)& -2-9(-2)& 0-9(-1)\\ -2+3\left(5\right)& 1+3(-2) & 0+3(-1)\\ 5& -2& -1\end{array} \right]$

$\left[\overline{)\mathrm{A} } \mathrm{I} \right] ~ \left[\overline{)\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array} } \begin{array}{ccc}-40& 16& 9\\ 13& -5 & -3\\ 5& -2& -1\end{array} \right]$

$\left[\overline{)\mathrm{I} } {\mathrm{A}}^{-1} \right]=\left[\overline{)\mathrm{I} } {\mathrm{A}}^{-1} \right]$

$\mathrm{So}, {\mathrm{A}}^{-1}=\left[\begin{array}{ccc}-40& 16& 9\\ 13& -5& -3\\ 5& -2& -1\end{array}\right]$