JEE MAIN - Mathematics (2024 - 6th April Evening Shift - No. 6)
If $$A$$ is a square matrix of order 3 such that $$\operatorname{det}(A)=3$$ and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$$, then $$\mathrm{m}+2 \mathrm{n}$$ is equal to :
2
4
3
6
Explanation
$$\begin{aligned} & |A|=3 \\ & \left|\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}(2 A)^{-1}\right)\right)\right)\right| \\ & =\left|-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 A)^{-1}\right)\right)\right)\right|^2 \end{aligned}$$
$$\begin{aligned} & =4^6\left|-3 \operatorname{adj}\left(\operatorname{aadj}\left((2 A)^{-1}\right)\right)\right|^4 \\ & =4^6 \cdot 3^{12}\left|\operatorname{aadj}\left((2 A)^{-1}\right)\right|^8 \\ & =4^6 \cdot 3^{12} \cdot 3^{24}\left|(2 A)^{-1}\right|^{16} \\ & =4^6 \cdot 3^{36} 2^{-48}\left|A^{-1}\right|^{16} \\ & =\frac{2^{-36} 3^{36}}{3^{16}}=2^{-36} 3^{20} \\ & m=-36 \\ & n=20 \\ & m+2 n=4 \end{aligned}$$
Comments (0)
