Let A, B, C be 3 × 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements
(S1) A¹³ B²⁶ - B²⁶ A¹³ is symmetric
(S2) A²⁶ C¹³ - C¹³ A²⁶ is symmetric
Then,

Solution

Maths sir · 2 steps

Reduce each power to symmetric or skew first, then take the transpose of the commutator.

1. Classify each power. Since (Xⁿ)ᵀ = (Xᵀ)ⁿ, A¹³ and A²⁶ stay symmetric. For skew B, (B²⁶)ᵀ = (-B)²⁶ = B²⁶, so B²⁶ is symmetric. For skew C, (C¹³)ᵀ = (-C)¹³ = -C¹³, so C¹³ is skew-symmetric.

   (PQ - QP)ᵀ = QᵀPᵀ - PᵀQᵀ

   Statement	Types of P and Q	Transpose of PQ - QP	Result
   S1	P = A¹³ symmetric, Q = B²⁶ symmetric	QP - PQ = -(PQ - QP)	skew-symmetric, so S1 is false
   S2	P = A²⁶ symmetric, Q = C¹³ skew	(-Q)P - P(-Q) = PQ - QP	symmetric, so S2 is true

2. So. Only S2 is true, which is option (a).

The idea to keep. Even powers of a skew-symmetric matrix are symmetric and odd powers stay skew. Then PQ - QP is skew when P and Q are the same type, and symmetric when they are of opposite types.

Answer from the NTA final answer key. Solution by PrepWiser.