Let A and B be any two 3 × 3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Solution

Maths sir · 4 steps

Take the transpose of each expression, using Aᵀ = A and Bᵀ = -B, and see which one fails its claim.

1. The two rules. Symmetric means Aᵀ = A and skew-symmetric means Bᵀ = -B. Also (XY)ᵀ = Yᵀ Xᵀ, so the order flips, and (Bⁿ)ᵀ = (Bᵀ)ⁿ = (-1)ⁿ Bⁿ.
2. Powers of B. An even power of B is symmetric, since (B⁴)ᵀ = (-B)⁴ = B⁴. An odd power stays skew, since (B⁵)ᵀ = (-B)⁵ = -B⁵. Every power of A is symmetric.

   Option	Transpose	Claim holds?
   (a) A⁴ - B⁴	A⁴ - B⁴, the same matrix	symmetric, true
   (b) AB - BA	Bᵀ Aᵀ - Aᵀ Bᵀ = -BA + AB	symmetric, true
   (c) B⁵ - A⁵	-B⁵ - A⁵ = -(B⁵ + A⁵)	not -(B⁵ - A⁵), false
   (d) AB + BA	-BA - AB = -(AB + BA)	skew-symmetric, true

3. Why (c) fails. In B⁵ - A⁵ the part B⁵ is skew but A⁵ is symmetric. A skew part plus a symmetric part is skew only when the symmetric part is zero, and that need not happen for an arbitrary A.
4. So. The statement that is NOT true is (c): B⁵ - A⁵ is not skew-symmetric in general.

The idea to keep. For symmetric and skew matrices, transpose the expression: reverse every product and swap Bᵀ for -B. Odd powers of a skew matrix stay skew and even powers turn symmetric.

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