JEE Main 2025 — 2 April, Morning Shift — Question 33

Let $|A|=0 \Rightarrow \alpha \beta-(-6)=0 \Rightarrow \alpha \beta=-6$ and $\alpha+\beta=1 \Rightarrow \alpha \beta$ are roots of the equation

$x^{2}-x-6=0 \Rightarrow x=3,-2$. Since $\alpha>0$

$\Rightarrow \alpha=3, \beta=-2$

$\Rightarrow A=\left[\begin{array}{ll}3 & -1\\ 6 & -2\end{array}\right] \Rightarrow I+A=\left[\begin{array}{ll}4 & -1\\ 6 & -1\end{array}\right]$

$(I+A)^{2}=\left[\begin{array}{ll}10 & -3\\ 18 & -5\end{array}\right] \Rightarrow(I+A)^{4}=\left[\begin{array}{ll}46 & -15\\ 90 & -29\end{array}\right]$

$\Rightarrow(I+A)^{8}=\left[\begin{array}{cc}766 & -255\\ 1530 & -509\end{array}\right]$