If $A$ is a square matrix of order 3 such that $\operatorname{det}(A)=3$ and…

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 :
  1. 2
  2. 3
  3. 6
  4. 4

Solution

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

Asked in: JEE Main 2024 (06 Apr Shift 2)

Practice more Matrices questions on Aicharya