JEE Main 2025 — Mathematics Algebra

$$\begin{aligned} & |2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))| \\ & 2^3 \cdot|3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A})|^2 \\ & 2^3 \cdot\left(3^3\right)^2 \cdot|\mathrm{~A}|^2 \cdot|\operatorname{adj}(2 \mathrm{~A})|^2 \\ & 2^3 \cdot 3^6 \cdot|\mathrm{~A}|^2 \cdot\left(|2 \mathrm{~A}|^2\right)^2 \\ & 2^3 \cdot 3^6 \cdot|\mathrm{~A}|^2\left[\left(2^3\right)^2 \cdot|\mathrm{~A}|^2\right]^2 \\ & 2^3 \cdot 3^6 \cdot|\mathrm{~A}|^2 \cdot 2^{12} \cdot|\mathrm{~A}|^4 \\ & 2^{15} \cdot 3^6 \cdot|\mathrm{~A}|^6 \\ & 2^{15} \cdot 3^6 \cdot 5^6=2^\alpha \cdot 3^\beta \cdot 5^\gamma \\ & \alpha=15, \quad \beta=6, \quad \gamma=6 \\ & \alpha+\beta+\gamma=27\end{aligned}$$​∣2adj(3 Aadj(2 A))∣23⋅∣3 Aadj(2 A)∣223⋅(33)2⋅∣ A∣2⋅∣adj(2 A)∣223⋅36⋅∣ A∣2⋅(∣2 A∣2)223⋅36⋅∣ A∣2[(23)2⋅∣ A∣2]223⋅36⋅∣ A∣2⋅212⋅∣ A∣4215⋅36⋅∣ A∣6215⋅36⋅56=2α⋅3β⋅5γα=15,β=6,γ=6α+β+γ=27​