QJEEM24Mad69b: JEE Main 2024 Mathematics - Class 12 JEEM24Mad69b Solution: Let and . If , then is equal to:

$\begin{aligned}& \left|\operatorname{adj}\left(A-2 A^T\right) \cdot \operatorname{adj}\left(2 A-A^T\right)\right|=2^8 \\& P=A-2 A^{\top} \\& Q=2 A^T-A \Rightarrow Q^T=2 A^T-A=-P \\& |\operatorname{adj}(P) \operatorname{adj}(Q)| \Rightarrow|P Q|=-2^4 \\& \Rightarrow|P|(-|P|)=-2^4 \Rightarrow|P|=4 \text { and }|Q|=-4 \\& \left|A-2 A^T\right|=4 \\& A-2 A^T=\left[\begin{array}{lll}1 & 2 & \alpha \\1 & 0 & 1 \\0 & 1 & 2\end{array}\right]-2\left[\begin{array}{lll}1 & 1 & 0 \\2 & 0 & 1 \\\alpha & 1 & 2\end{array}\right]=\left[\begin{array}{ccc}-1 & 0 & \alpha \\-3 & 0 & -1 \\-2 \alpha & -1 & -2\end{array}\right] \\& \Rightarrow\left|A-2 A^T\right|=1+3 \alpha=4 \Rightarrow \alpha=1 \Rightarrow|A|=-4 \\& \Rightarrow|A|^2=16\end{aligned}$