Question

Let $A$ be a $2\times 2$ matrix with det $\left(A\right)=-1$ and det $\left(\right(A+I\left)\right($Adj $\left(A\right)+I\left)\right)=4$. Then the sum of the diagonal elements of $A$ can be

Answer 1

$\left|\right(A+I\left)\right(\text{adj}A+I\left)\right|=4$

$\Rightarrow |A\text{adj}A+A+\text{adj}A+I|=4$

$\Rightarrow \left|\right(A)|I+A+\text{adj}A+I|=4$

$\left|A\right|=–1\Rightarrow |A+\text{adj}A|=4$

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{adj}A=\left[\begin{array}{cc}a& -b\\ -c& d\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}(a+d)& 0\\ 0& (a+d)\end{array}\right]=4$

$\Rightarrow a+d=\pm 2$