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 m n , m n . Then 4 m +2 n is equal to \ \ \ \ \ .

Mathematics · Matrices

JEE Main 2025 — 22 January, Morning Shift — Question 23

Let $A$ be a square matrix of order 3 such that $\operatorname{det}(A)=-2$ and $\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=2^{m+n} \cdot 3^{m n}$ , $\mathrm{m}>\mathrm{n}$ . Then $4 \mathrm{~m}+2 \mathrm{n}$ is equal to $\_\_\_\_\_$ .

Answer: 34

Numerical answer — enter this value.

Step-by-step solution

$|\mathrm{A}|=-2$

$\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))$

$=3^{3} \operatorname{det}(\operatorname{adj}(-6\operatorname{adj}(3 A)))$

$=3^{3}(-6)^{6}(\operatorname{det}(3 A))^{4}$

$=3^{21} \times 2^{10}$

$\mathrm{m}+\mathrm{n}=10$

$\mathrm{mn}=21$

$\mathrm{m}=7 ; \mathrm{n}=3$

$4 \mathrm{~m}+2 \mathrm{n}=34$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 22 January, Morning Shift
Subject: Mathematics
Chapter: Matrices
Topic: Adjoint of a Square Matrix