JEE 2023 — Mathematics PYQ

Given $A=\left[\begin{array}{ccc}2& amp;1& amp;0\\ 1& amp;2& amp;-1\\ 0& amp;-1& amp;2\end{array}\right]$, $|A|=2(4-1)-1(2-0)=4$
$|2A|=2^3|A|=8\times 4=32$
Property: $|\text{adj}(\text{adj}(\text{adj }M))|=|M{|}^{(n-1{)}^3}$
$\Rightarrow |\text{adj}(\text{adj}(\text{adj }2A))|=|2A{|}^{(3-1{)}^3}=(32{)}^{2^3}=32^8$
Given $16^n=(2^5{)}^8=2^{40}$
$\Rightarrow (2^4{)}^n=2^{40}\Rightarrow 4n=40\Rightarrow n=10$