If for the matrix, A = [ 1, - α; α, β ], AAᵀ = I₂, then the value of α ⁴ + β ⁴ is :

Solution

Maths sir · 3 steps

Multiply out AAᵀ and match it with I₂ entry by entry. The top-left entry alone forces α = 0.

1. Write the transpose. Swapping rows and columns of A gives Aᵀ = [1, α; -α, β].

   AAᵀ = [1+α², α-αβ; α-αβ, α²+β²] = [1, 0; 0, 1]

2. Compare entries. Top-left: 1 + α² = 1, so α = 0. The off-diagonal entry α(1-β) = 0 then holds automatically. Bottom-right: α² + β² = 1 gives β² = 1.
3. So. α⁴ + β⁴ = 0 + (β²)² = 0 + 1 = 1, which is option (c).

The idea to keep. For an orthogonal matrix, AAᵀ = I means each row has length 1 and different rows are perpendicular. Start with the diagonal entries: they usually pin down the unknowns fastest.

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