For a 3 × 3 matrix M , let trace (M) denote the sum of all the diagonal elements of M . Let A be a 3 × 3 matrix such that A =1/2 and trace ( A )=3 . If B= adj ( adj (2 A)) , then the value of B + trace (B) equals:

Mathematics · Matrices

JEE Main 2025 — 22 January, Evening Shift — Question 4

For a $3 \times 3$ matrix $M$ , let trace $(M)$ denote the sum of all the diagonal elements of M . Let A be a $3 \times 3$ matrix such that $|\mathrm{A}|=\frac{1}{2}$ and trace $(\mathrm{A})=3$ . If $B=\operatorname{adj}(\operatorname{adj}(2 A))$ , then the value of $|B|+$ trace $(B)$ equals:

Option A:
56
Option B:
132
Option C:
174
Option D:
280
Correct

Answer: D

Step-by-step solution

$|\mathrm{A}|=\frac{1}{2}, \operatorname{trace}(\mathrm{~A})=3, \mathrm{~B}=\operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))=|2 \mathrm{~A}|^{n-2}(2 \mathrm{~A})$

$\mathrm{n}=3, \mathrm{~B}=|2 \mathrm{~A}|(2 \mathrm{~A})=2^{3} .|\mathrm{A}|(2 \mathrm{~A})=8 \mathrm{~A}$

$|\mathrm{B}|=|8 \mathrm{~A}|=8^{3} .|\mathrm{A}|=2^{8}=256$

$\operatorname{trace}(\mathrm{B})=8 \operatorname{trace}(\mathrm{~A})=24$

$|\mathrm{B}|+\operatorname{trace}(\mathrm{B})=280$

Answer key and solution verified before publishing.

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

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Answer: D

Step-by-step solution

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