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JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Matrices

Let

A = 210 12 - 1 0 - 1 2

. If

∣ adj (adj (adj 2 A)) ∣ = (16)^{n}

, then n is equal to

(A)10✓ Correct
(B)9
(C)12
(D)8

Answer: (A)

Solution

For a

k \times k

matrix

M

,

∣ adj (M) ∣ = ∣ M ∣^{k - 1}

.

For matrix

A

of order 3:

∣2 A ∣ = 2^{3} ∣ A ∣ = 8∣ A ∣

.

Compute

∣ A ∣

: expanding along first row gives

∣ A ∣ = 2 (4 - 1) - 1 (2 - 0) = 6 - 2 = 4

.

So

∣2 A ∣ = 8 \times 4 = 32 = 2^{5}

.

Applying the adjugate property three times:

∣ adj (adj (adj (2 A))) ∣ = ∣2 A ∣^{(3 - 1)^{3}} = ∣2 A ∣^{8} = (2^{5})^{8} = 2^{40} = (2^{4})^{10} = 1 6^{10}

.

Therefore

n = 10

.

JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Single correct option

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