Mathematics-I (3110014)

$\mathrm{Now}, \mathrm{A}-\mathrm{\lambda I}=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]-\mathrm{\lambda } \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\mathrm{Now}, \mathrm{A}-\mathrm{\lambda I}=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]- \left[\begin{array}{cc}\mathrm{\lambda }& 0\\ 0& \mathrm{\lambda }\end{array}\right]$

$\mathrm{Now}, \mathrm{A}-\mathrm{\lambda I}=\left[\begin{array}{cc}1-\mathrm{\lambda }& -1\\ 2& 3-\mathrm{\lambda }\end{array}\right]$

$\mathrm{Characteristic} \mathrm{equation} \mathrm{is},$

$|\mathrm{A}-\mathrm{\lambda I}|=0$

$\Rightarrow \left|\begin{array}{cc}1-\mathrm{\lambda }& -1\\ 2& 3-\mathrm{\lambda }\end{array}\right|=0$

$\Rightarrow (1-\mathrm{\lambda })(3-\mathrm{\lambda })-(-1)\left(2\right)=0$

$\Rightarrow 3-4\mathrm{\lambda }+{\mathrm{\lambda }}^2+2=0$

$\Rightarrow {\mathrm{\lambda }}^2-4\mathrm{\lambda }+5=0$

$\mathrm{By} \mathrm{Cayley}-\mathrm{Hamilton} \mathrm{Theorem},$

${\mathrm{A}}^2-4\mathrm{A}+5\mathrm{I}=0$

$⟹{\mathrm{A}}^{-1} {\mathrm{A}}^2-4{\mathrm{A}}^{-1} \mathrm{A}+5{\mathrm{A}}^{-1} \mathrm{I}=0{\mathrm{A}}^{(}-1) [ \mathrm{Multiplied} \mathrm{by} {\mathrm{A}}^{-1} ]$

$⟹\mathrm{A}-4\mathrm{I}+5{\mathrm{A}}^{-1}=0$

$⟹5{\mathrm{A}}^{-1}=4\mathrm{I}-\mathrm{A}$

$⟹5{\mathrm{A}}^{-1} =4 \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] - \left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$

$⟹5{\mathrm{A}}^{-1} = \left[\begin{array}{cc}4& 0\\ 0& 4\end{array}\right] - \left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$

$⟹5{\mathrm{A}}^{-1} = \left[\begin{array}{cc}4-1& 0+1\\ 0-2& 4-3\end{array}\right]$

$⟹{\mathrm{A}}^{-1} = \frac{1}{5}\left[\begin{array}{cc} 3& 1\\ -2 & 1\end{array}\right]$