JEE Main 2026 — Jan 22, Shift 1 — Mathematics solution

JEE Main2026Jan 22, Shift 1Mathematics

Let A be a $3 \times 3$ matrix such that $A + A^{T} = O$ . If $A 1 - 1 0 = 332, A^{2} 1 - 1 0 = - 3 19 - 24$ and $det (adj (2 adj (A + I))) = (2)^{α} \cdot (3)^{β} \cdot (11)^{γ}$ , $α, β, γ$ are non-negative integers, then $α + β + γ$ is equal to _____.

Your answer

Solution

$A + A^{T} = O$ means $A$ is skew-symmetric: $A = 0 ab - a 0 c - b - c 0$ .

From $A 1 - 1 0 = 332$ : $- a = 3$ (twice), $c - b = 2$ , so $a = - 3$ .

From $A^{2} v = A 332 = - 3 19 - 24$ : computing yields $b = 3$ , $c = 5$ (consistency check passes).

So $A + I = 1 - 33 315 - 3 - 51$ , with $∣ A + I ∣ = 1 (1 + 25) + 3 (3 + 15) + 3 (- 15 + 3) = 26 + 54 - 36 = 44$ .

Use the determinant identities for $n = 3$ : $∣ a d j (M) ∣ = ∣ M ∣^{2}$ and $∣ c M ∣ = c^{3} ∣ M ∣$ .

$∣2 adj (A + I) ∣ = 2^{3} ∣ a d j (A + I) ∣ = 8 \cdot 4 4^{2}$ . Then $∣ adj (2 adj (A + I)) ∣ = (8 \cdot 4 4^{2})^{2} = 64 \cdot 4 4^{4} = 2^{6} \cdot (2^{2} \cdot 11)^{4} = 2^{14} \cdot 1 1^{4}$ .

So $α = 14, β = 0, γ = 4$ , and $α + β + γ = 18$ .

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