If A = [ cos θ, i sin θ; i sin θ, cos θ ], ( θ = π/24 )
and A⁵ = [ a, b; c, d ], where i = √(- 1) then which one of the following is not true?

Solution

Maths sir · 3 steps

Powering this matrix just multiplies the angle, so A⁵ has 5θ wherever A had θ.

1. Square it first. Multiplying A by itself gives diagonal entries cos² θ - sin² θ = cos 2θ and off-diagonal entries 2i sin θ cos θ = i sin 2θ. So A² has the same shape as A, with θ replaced by 2θ. The same multiplication repeated gives the pattern for every power.

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

2. Read off the entries. With n = 5 and φ = 5θ = 5π/24, we get a = d = cos φ and b = c = i sin φ. Because i² = -1, b² = c² = -sin² φ.

   Option	Works out to	True?
   (a) a² - c²	cos² φ + sin² φ = 1	true
   (b) a² + b²	cos² φ - sin² φ = cos 5π/12 ≈ 0.26	true, it lies in [0, 1]
   (c) a² - d²	cos² φ - cos² φ = 0	true
   (d) a² - b²	cos² φ + sin² φ = 1	false, it is not 1/2

3. So. a² - b² = 1, not 1/2. The statement that is not true is (d).

The idea to keep. If A² keeps the same shape as A, find the pattern and use it for Aⁿ. Then watch the sign: (i sin φ)² = -sin² φ, so subtracting it adds sin² φ.

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