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JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Matrices

Let

∣ A ∣ = 6

, where A is a

3 \times 3

matrix. If

∣ adj (3 adj (A^{2} . adj (2 A))) ∣ = 2^{m} \cdot 3^{n}, m, n \in N

, then

m + n

is equal to ______.

Answer: 62

Solution

Using

adj (k A) = k^{n - 1} adj (A)

for

n \times n

matrix:

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

.

A^{2} (adj (2 A)) = 4 A^{2} adj (A) = 4 A (A adj (A)) = 4 A ∣ A ∣ I = 24 A I

.

So

A^{2} adj (2 A) = 24 A

.

3 adj (A^{2} adj (2 A)) = 3 adj (24 A) = 3 \cdot 2 4^{2} adj (A) = 3 \cdot 2^{6} \cdot 3^{2} adj (A) = 2^{6} \cdot 3^{3} adj (A)

.

∣ adj (3 adj (A^{2} adj (2 A))) ∣ = ∣ (2^{6} \cdot 3^{3})^{2} adj (adj (A)) ∣ = (2^{12} \cdot 3^{6})^{3} ∣ A ∣^{4} = 2^{36} \cdot 3^{18} \cdot 6^{4} = 2^{36} \cdot 3^{18} \cdot 2^{4} \cdot 3^{4} = 2^{40} \cdot 3^{22}

.

Thus

m = 40, n = 22

, so

m + n = 62

.

JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Integer answer

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