Let A be a non-singular matrix of order 3 . If det(3 adj(2 adj((det A) A)))=3^ -13 · 2^ -10 and det(3 adj(2 ~A))=2^ m · 3^ n , then |3 ~m+2 n| is equal to

Correct answer: 0 aligned & |3 adj(2 adj(|A| A))|= 3 adj(2|A|^2 adj(A) . & =.|3.2^2| A|^4 adj(.adj(A)|=2^6 3^3| A|^ 12 |A|^4. & =2^6 3^3|A|^ 16 =2^ -10 3^ -13 & |A|^ 16 =2^ -16 3^ -16 |A|=2^ -1 3^ -1 aligned aligned & Now |3 adj(2 A)|=|^ 3 .2^2 adj(A)| & =2^6 3^3| A|^2=2^ -m 3^ -n & 2^6 3^3 2^ -2 3^ -2 =2^ -m 3^ -n & 2^ -m 3^ -n =2^4 3^1 & m=-4, n=-1 & |3 m+2 n|=|-12-2|=14 aligned

JEE Main 20249 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 9 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2024 · 9 April · Morning Shift · Mathematics · Matrices

Let A be a non-singular matrix of order 3 . If det(3 adj(2 adj((det...

Let

A

be a non-singular matrix of order 3 . If

\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}

and

\operatorname{det}(3 \operatorname{adj}(2 \mathrm{~A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}

, then

|3 \mathrm{~m}+2 \mathrm{n}|

is equal to

\qquad

Correct answer

Step-by-step solution

\begin{aligned} & |3 \operatorname{adj}(2 \operatorname{adj}(|A| A))|=\mid 3 \operatorname{adj}\left(2|A|^2 \operatorname{adj}(A) \mid\right. \\ & =\left.\left|3.2^2\right| A\right|^4 \operatorname{adj}\left(\left.\operatorname{adj}(A)\left|=2^6 3^3\right| A\right|^{12}|A|^4\right. \\ & =2^6 3^3|A|^{16}=2^{-10} 3^{-13} \\ & \Rightarrow|A|^{16}=2^{-16} 3^{-16} \Rightarrow|A|=2^{-1} 3^{-1}\end{aligned}

\begin{aligned} & \text{ Now }|3 \operatorname{adj}(2 \mathrm{ A})|=\left|^{3}.2^2 \operatorname{adj}(\mathrm{A})\right| \\ & =2^6 3^3|\mathrm{ A}|^2=2^{-\mathrm{m}} 3^{-\mathrm{n}} \\ & \Rightarrow 2^6 3^3 2^{-2} 3^{-2}=2^{-\mathrm{m}} 3^{-\mathrm{n}} \\ & \Rightarrow 2^{-\mathrm{m}} 3^{-\mathrm{n}}=2^4 3^1 \\ & \Rightarrow \mathrm{m}=-4, \mathrm{n}=-1 \\ & \Rightarrow|3 \mathrm{ m}+2 \mathrm{n}|=|-12-2|=14\end{aligned}

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