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JEE Main 202522 JanuaryEvening Shift

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

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

For a 3 × 3 matrix M , let trace (M) denote the sum of all the...

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

|A|=\frac{1}{2}

and trace

(A)=3

. If

B=\operatorname{adj}(\operatorname{adj}(2 A))

, then the value of

|B|+

trace (B) equals :

Options

A56
B132
C174
D280

Correct answer

D. 280

Step-by-step solution

\because \operatorname{tr}(A)=3 \text{

and

}|A|=\frac{1}{2}

Now,

B=\operatorname{adj}(\operatorname{adj}(2 A))=|2 A|^{3-2} \cdot(2 A) =2^3|A| \cdot 2 A=8 A

\begin{aligned} & \therefore \operatorname{tr}(B)=8 \operatorname{tr}(A)=24 \\ & \text{ and }|B|=|^{8} A|=8^3 \cdot \frac{1}{2}=256 \\ & \therefore \quad \operatorname{trace}(B)+|B|=24+256\end{aligned}

=280

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