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JEE Main · 2026 · 22 Jan 2026 · Shift 1 · Matrices

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}

, where

α, β, γ

are non-negative integers, then

α + β + γ

is equal to

Answer: 18

Solution

Since

A + A^{T} = O

,

A

is skew-symmetric; using the two given image vectors it is

A = 03 - 3 - 3 0 - 5 350

. Then

A + I = 13 - 3 - 3 1 - 5 351

with

det (A + I) = 26 + 54 - 36 = 44 = 2^{2} \cdot 11

. For a

3 \times 3

matrix

det (adj M) = (det M)^{2}

and

det (k M) = k^{3} det M

. Writing

B = A + I

:

det (adj B) = 4 4^{2}

,

det (2 adj B) = 2^{3} \cdot 4 4^{2}

, and

det (adj (2 adj B)) = (2^{3} \cdot 4 4^{2})^{2} = 2^{6} \cdot 4 4^{4} = 2^{14} \cdot 3^{0} \cdot 1 1^{4}

. Hence

α = 14, β = 0, γ = 4

and

α + β + γ = 18

.

JEE Main · 2026 · 22 Jan 2026 · Shift 1 · Integer answer

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