Question

Let A = `[(cosα, -sinα),(sinα, cosα)]`, α ∈ R such that A³² = `[(0, -1),(1, 0)]`. Then a value of α is ______.

विकल्प

• 0

• `π/16`

• `π/32`

• `π/64`

Answer 1

Let A = `[(cosα, -sinα),(sinα, cosα)]`, α ∈ R such that A³² = `[(0, -1),(1, 0)]`. Then a value of α is `underlinebb(π/64)`.

Explanation:

A = `[(cosα, -sinα),(sinα, cosα)]`

A² = `[(cosα, -sinα),(sinα, cosα)][(cosα, -sinα),(sinα, cosα)]`

= `[(cos^2α - sin^2α, -2cosαsinα),(2sinαcosα, cos^2α - sin^2α)]`

= `[(cos2α, -sin2α),(sin2α, cos2α)]`

Similarly, 

A³² = `[(cos32α, -sin32α),(sin32α, cos32α)] = [(0, -1),(1, 0)]`

⇒ cos32α = 0 and sin32α = 1

32α = `π/2`, 32α = `π/2`

α = `π/64`, α = `π/64`