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JEE Main 2023 · 25 Jan · Shift 2

JEE Main 2023 Mathematics question on Matrices and Determinants

Let A, B, C be 3×33 \times 3 matrices such that A is symmetric and B and C are skew-symmetric.

Consider the statements

(S1) A13B26−B26A13A^{13} B^{26} - B^{26} A^{13} is symmetric

(S2) A26C13−C13A26A^{26} C^{13} - C^{13} A^{26} is symmetric

Then,
  1. A.
    Only S2 is true
  2. B.
    Both S1 and S2 are false
  3. C.
    Both S1 and S2 are true
  4. D.
    Only S1 is true
Show answer and solution

Correct option: A

Solution

For any matrix M, if M is symmetric (Mᵀ = M) or skew-symmetric (Mᵀ = -M), then (Mⁿ)ᵀ = (Mᵀ)ⁿ.

Key facts:
- A symmetric: Aᵀ = A, so (An)ᵀ = Aⁿ (always symmetric for any n)
- B skew-symmetric: Bᵀ = -B, so (Bn)ᵀ = (-1)ⁿ Bⁿ. For B²⁶: (B²⁶)ᵀ = B²⁶ (symmetric, even power)
- C skew-symmetric: (C¹³)ᵀ = (-1)¹³ C¹³ = -C¹³ (skew-symmetric, odd power)

For S1: Let P = A¹³B²⁶ - B²⁶A¹³. Take transpose:
Pᵀ = (B²⁶)ᵀ(A¹³)ᵀ - (A¹³)ᵀ(B²⁶)ᵀ = B²⁶·A¹³ - A¹³·B²⁶ = -P.
So P is skew-symmetric, not symmetric. S1 is false.

For S2: Let Q = A²⁶C¹³ - C¹³A²⁶. Take transpose

[JEE Main 2023 · 25 Jan · Shift 2]

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