13. Let A be an n x n (real) matrix with characteristic roots , , ..., , where any , may not be a real

13. Let A be an n x n (real) matrix with characteristic roots λ₁, λ₂, ..., λₙ, where any

λᵢ, may not be a real number. Denote λᵢ by xᵢ + iyᵢ, where xᵢ and yᵢ are real numbers and where i = √-1. Show that:

(a) Σ yᵢ = 0.

(b) Σ xᵢyᵢ = 0.

(c) tr (A²) = Σ xᵢ² - Σ yᵢ².