Let A be a square matrix of order 3 such that det(A)=-2 and det(3 adj(-6 adj(3 A)))=2^ m+n · 3^ mn , m . Then 4 ~m+2 n is equal to _______

Correct answer: 0 aligned & As A adj A=|A| l, det(λ A)=λ^n det A & det(3 adj(-6 adj(3 A)))=3^3 det(adj(-6 adj(3 A))) & =3^3(-6 adj(3 A))^2 & =3^3(-6)^6|3 A|^4 & =3^9 2^6 · 3^ 12 ·(-2)^4 & =3^ 21 · 2^ 10 aligned Now comparing with given condition aligned & 2^ m+n 3^ m n =2^ 10 · 3^ 21 & m+n=10, m n=21 & m=7, n=3(m>n) & 4 m+2 n=28+6=34 aligned

JEE Main 202522 JanuaryMorning Shift

Exam: JEE Main
Year: 2025
Date: 22 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2025 · 22 January · Morning Shift · Mathematics · Matrices

Let A be a square matrix of order 3 such that det(A)=-2 and det(3...

Let

A

be a square matrix of order 3 such that

\operatorname{det}(A)=-2

and

\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=2^{\mathrm{m}+\mathrm{n}} \cdot 3^{\mathrm{mn}}, \mathrm{m}\gt\mathrm{n}

. Then

4 \mathrm{~m}+2 \mathrm{n}

is equal to _______

Correct answer

Step-by-step solution

\begin{aligned} & \text{ As } A \operatorname{adj} A=|A| l, \operatorname{det}(\lambda A)=\lambda^n \operatorname{det} A \\ & \operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=3^3 \operatorname{det}(\operatorname{adj}(-6 \operatorname{adj}(3 A))) \\ & =3^3(-6 \operatorname{adj}(3 A))^2 \\ & =3^3(-6)^6|3 A|^4 \\ & =3^9 2^6 \cdot 3^{12} \cdot(-2)^4 \\ & =3^{21} \cdot 2^{10} \end{aligned}

Now comparing with given condition

\begin{aligned} & 2^{m+n} 3^{m n}=2^{10} \cdot 3^{21} \\ & m+n=10, m n=21 \\ & \Rightarrow \quad m=7, n=3(m>n) \\ & \therefore \quad 4 m+2 n=28+6=34 \end{aligned}

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