Mathematics - Matrices Question with Solution | TestHub

$\mathrm{I}+\mathrm{A}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\left(\begin{array}{cc}1& 2\\ 1& 1\end{array}\right)=\left(\begin{array}{cc}2& 2\\ 1& 2\end{array}\right)$

and $\mathrm{I}-\mathrm{A}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-\left(\begin{array}{cc}1& 2\\ 1& 1\end{array}\right)=\left(\begin{array}{cc}0& -2\\ -1& 0\end{array}\right)$

Now, $|\mathrm{I}-\mathrm{A}|=\left|\begin{array}{cc}0& -2\\ -1& 0\end{array}\right|=0-2=-2$.

$\mathrm{adj}(\mathrm{I}-\mathrm{A})=\left(\begin{array}{cc}0& 2\\ 1& 0\end{array}\right)$.

$(\mathrm{I}-\mathrm{A}{)}^{-1}=-\frac{1}{2}\left(\begin{array}{cc}0& 2\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}0& -1\\ -\frac{1}{2}& 0\end{array}\right)$

$$\varphi (\mathrm{A})=(\mathrm{I}+\mathrm{A})(\mathrm{I}-\mathrm{A}{)}^{-1}=\left(\begin{array}{cc}2& 2\\ 1& 2\end{array}\right)\left(\begin{array}{cc}0& -1\\ -\frac{1}{2}& 0\end{array}\right)=\left(\begin{array}{cc}-1& -2\\ -1& -1\end{array}\right)=-\mathrm{A}$$