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Determinants · Question 18

JEE Main 2026 (Online) 22nd January Morning ShiftNumericalHard📈 Topic in 42% papers+4
1

Let A be a 3 3 matrix such that A + AT = O. If A bmatrix 1 \ -1 \ 0 bmatrix = bmatrix 3 \ 3 \ 2 bmatrix, A2 bmatrix 1 \ -1 \ 0 bmatrix = bmatrix -3 \ 19 \ -24 bmatrix and (adj(2 adj(A + I))) = (2) (3) (11) , , , are non-negative integers, then + + is equal to

Answer

18

Solution

A is skew-symmetric: let A = bmatrix 0 & p & q \ -p & 0 & r \ -q & -r & 0 bmatrix

A bmatrix 1 \ -1 \ 0 bmatrix = bmatrix -p \ -p \ r - q bmatrix = bmatrix 3 \ 3 \ 2 bmatrix

p = -3, r - q = 2

A2 bmatrix 1 \ -1 \ 0 bmatrix = A bmatrix 3 \ 3 \ 2 bmatrix = bmatrix 3p + 2q \ -3p + 2r \ -3q - 3r bmatrix = bmatrix -3 \ 19 \ -24 bmatrix

-9 + 2q = -3 q = 3; 9 + 2r = 19 r = 5 (check: -3(3 + 5) = -24)

A + I = bmatrix 1 & -3 & 3 \ 3 & 1 & 5 \ -3 & -5 & 1 bmatrix

|A + I| = 1(1 + 25) + 3(3 + 15) + 3(-15 + 3)

= 26 + 54 - 36 = 44

For a matrix M of order 3: |adj M| = |M|2 and |kM| = k3|M|

Let B = 2 adj(A + I): |B| = 23 442

|adj B| = |B|2 = 26 444

= 26 28 114 = 214 30 114

= 14, = 0, = 4

+ + = 18

Answer: 18

JEE Main 2026 (Online) 22nd January Morning Shift

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