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JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Determinants

Let

x, y, z > 1

and

A = 1 lo g_{y} x lo g_{z} x lo g_{x} y 2 lo g_{z} y lo g_{x} z lo g_{y} z 3

.
Then

adj (adj A^{2})

is equal to

(A) $6^{4}$
(B) $2^{8}$ ✓ Correct
(C) $4^{8}$
(D) $2^{4}$

Answer: (B)

Solution

Expressing each logarithm over a common base and factoring, the determinant collapses to

∣ A ∣ = 2

. For a

3 \times 3

matrix

∣ adj (adj M) ∣ = ∣ M ∣^{(3 - 1)^{2}} = ∣ M ∣^{4}

, so

adj (adj A^{2}) = ∣ A^{2} ∣^{4} = (2^{2})^{4} = 2^{8}

.

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Single correct option

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