QJEEM24Ma884a: JEE Main 2024 Mathematics - Class 12 JEEM24Ma884a Solution: If is a square matrix of order 3 such that and , then is equal…

$\begin{aligned}& |A|=3 \\& \left|\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}(2 A)^{-1}\right)\right)\right)\right| \\& =\left|-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 A)^{-1}\right)\right)\right)\right|^2\end{aligned}$

$\begin{aligned}& =4^6\left|-3 \operatorname{adj}\left(\operatorname{aadj}\left((2 A)^{-1}\right)\right)\right|^4 \\& =4^6 \cdot 3^{12}\left|\operatorname{aadj}\left((2 A)^{-1}\right)\right|^8 \\& =4^6 \cdot 3^{12} \cdot 3^{24}\left|(2 A)^{-1}\right|^{16} \\& =4^6 \cdot 3^{36} 2^{-48}\left|A^{-1}\right|^{16} \\& =\frac{2^{-36} 3^{36}}{3^{16}}=2^{-36} 3^{20} \\& m=-36 \\& n=20 \\& m+2 n=4\end{aligned}$