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JEE Main · 2025 · 22 Jan 2025 · Shift 2 · Matrices

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 = adj (adj (2 A))

, then the value of

∣ B ∣ +

trace

(B)

equals:

(A)56
(B)132
(C)174
(D)280✓ Correct

Answer: (D)

Solution

For an

n \times n

matrix M,

adj (adj (M)) = ∣ M ∣^{n - 2} M

. Here

n = 3

, so

adj (adj (M)) = ∣ M ∣ \cdot M .

With

M = 2 A

:

∣2 A ∣ = 2^{3} ∣ A ∣ = 8 \cdot \frac{1}{2} = 4

, so

B = adj (adj (2 A)) = 4 \cdot (2 A) = 8 A .

Then

∣ B ∣ = ∣8 A ∣ = 8^{3} ∣ A ∣ = 512 \cdot \frac{1}{2} = 256, trace (B) = 8 trace (A) = 8 \cdot 3 = 24.

So

∣ B ∣ + trace (B) = 256 + 24 = 280.

JEE Main · 2025 · 22 Jan 2025 · Shift 2 · Single correct option

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