Question

If a² + b² + c² = 1, prove that |(a² + (b² + c²)cosϕ, ab(1 - cosϕ), ac(1 - cosϕ)), (ba(1 - cosϕ), b² + (c + a²)cosϕ, bc(1 - cosϕ)), (ca(1 - cosϕ), cb(1 - cosϕ), c² + (a² + b²)cosϕ)| is independent of a, b and c.

Answer 1

Multiplying C₁, C₂, C₃ by a, b, c, respectively, and taking a, b, c common from R₁, R₂, R₃, respectively, we get

 Taking a² + b² + c² common from C₁, we get

= (a² + b² + c²)² cos²ϕ {by property since all elements are zero below leading diagonal}

= 1² cos²ϕ = cos²ϕ, which is independent of a, b and c.