Let A be a square matrix such that AAᵀ=I. Then 1/2 A[(A+Aᵀ)²+(A-Aᵀ)²] is equal to

Solution

Maths sir · 4 steps

Expand both squares carefully: the cross terms cancel, and then AAᵀ = I does the rest.

1. Expand both squares. Matrices need not commute, so keep the order: (A+Aᵀ)² = A² + AAᵀ + AᵀA + (Aᵀ)² and (A-Aᵀ)² = A² - AAᵀ - AᵀA + (Aᵀ)².
2. Add them. The cross terms AAᵀ and AᵀA appear once with a plus sign and once with a minus sign, so they cancel. No commuting is needed for this.

   1/2A[2A² + 2(Aᵀ)²] = A³ + AAᵀAᵀ

3. Use the given condition. Group the last term as (AAᵀ)Aᵀ = I Aᵀ = Aᵀ.
4. So. The expression equals A³ + Aᵀ, which is option (c).

The idea to keep. For matrices, (X+Y)² + (X-Y)² = 2(X² + Y²) always holds, even when XY ≠ YX. Then group factors so that AAᵀ sits side by side and becomes I.

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