Question

Find A^–1 using adjoint method, where A = `[(cos theta, sin theta),(-sin theta, cos theta)]`

Answer 1

|A| = cos θ (cos θ) – sin θ (– sin θ)

= cos²θ + sin²θ

= 1 ≠ 0

∴ A^–1 exists.

A₁₁ = (–1)¹⁺¹ M₁₁ = M₁₁ = cos θ

A₁₂ = (–1)¹⁺² M₁₂ = – M₁₂ = sin θ

A₂₁ = (–1)²⁺¹ M₂₁ = – M₂₁ = – sin θ

A₂₂ = (–1)²⁺² M₂₂ = M₂₂ = cos θ

∴ adj (A) = `[(cos theta, sin theta),(-sin theta, cos theta)]^"T"`

= `[(cos theta, -sin theta),(sintheta, cos theta)]`

∴ A^–1 = `1/|"A"|` adj (A)

= `[(cos theta, -sin theta),(sin theta, cos theta)]`