JEE Main 2024 — 6 April, Shift 2 — Question 20

$|\mathrm{A}|=3$ $\left|\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right|$ $\mid-4 \operatorname{adj}\left(-\left.3 \operatorname{adj}\left(3 \operatorname{adj}(2 \mathrm{~A})^{-1}\right)\right|^{2}\right.$

$4^{6} \mid \operatorname{adj}\left(-\left.3 \operatorname{adj}\left(3 \operatorname{adj}(2 \mathrm{~A})^{-1}\right)\right|^{2}\right.$

$2^{12} \cdot 3^{12}\left|3 \operatorname{adj}(2 \mathrm{~A})^{-1}\right|^{8}$

$2^{12} \cdot 3^{12} \cdot 3^{24}\left|\operatorname{adj}(2 \mathrm{~A})^{-1}\right|^{8}$

$2^{12} \cdot 3^{36}\left|(2 \mathrm{~A})^{-1}\right|^{16}$

$2^{12} \cdot 3^{36} \frac{1}{|2 \mathrm{~A}|^{16}}$

$2^{12} \cdot 3^{36} \frac{1}{2^{48}|\mathrm{~A}|^{16}}$

$2^{12} \cdot 3^{36} \frac{1}{2^{48} \cdot 3^{16}}$ $\frac{3^{20}}{2^{36}}=2^{-36} \cdot 3^{20}$

$\mathrm{m}=-36 \quad \mathrm{n}=20$

$\mathrm{m}+2 \mathrm{n}=4$