If A = 1/2[ 1, √3; - √3, 1 ], then :

Solution

Maths sir · 4 steps

Read A as a rotation by 60°: six turns bring you back to I, so big powers collapse fast.

1. Spot the rotation. Since 1/2 = cos 60° and √3/2 = sin 60°, the matrix is A = (cos θ, sin θ; -sin θ, cos θ) with θ = 60°. When you multiply matrices of this form, their angles add.

   Aⁿ = (cos nθ, sin nθ; -sin nθ, cos nθ)

2. Find the cycle. With n = 6 the angle is 360°, so cos 360° = 1 and sin 360° = 0. That gives A⁶ = I.
3. Reduce the powers. A³⁰ = (A⁶)⁵ = I. Also A²⁵ = (A⁶)⁴ · A = A.
4. So. A³⁰ + A²⁵ - A = I + A - A = I, which is option (b). The other options reduce to I - A = 2I, I = A and I + 2A = I, and all three are false.

The idea to keep. When A is a rotation matrix by θ, Aⁿ is the rotation by nθ. Find the smallest n with Aⁿ = I, then cut every exponent down using that cycle.

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