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JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Matrices

Let A be a

3 \times 3

matrix such that

∣ adj (adj (adj A)) ∣ = 1 2^{4}

. Then

A^{- 1} adj A

is equal to

(A) $23$ ✓ Correct
(B) $6$
(C)12
(D)1

Answer: (A)

Solution

For a

3 \times 3

matrix,

∣ adj A ∣ = ∣ A ∣^{2}

, and each further adjugate squares the determinant again, so

∣ adj (adj (adj A)) ∣ = ∣ A ∣^{2^{3}} = ∣ A ∣^{8}

. From

∣ A ∣^{8} = 1 2^{4}

follows

∣ A ∣^{2} = 12

, so

∣ A ∣ = 23

. Then

A^{- 1} adj A = \frac{1}{∣ A ∣} \cdot ∣ A ∣^{2} = ∣ A ∣ = 23

.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Single correct option

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