Chapter 5, Problem 22SE

Exercises 19–23 concern the polynomial

p(t) = a₀ + a₁t + … + a_n−1tⁿ⁻¹ + tⁿ

and an n × n matrix C_p called the companion matrix of p:

C_p = $\left[\begin{array}{ccccc}\text{0}& \text{1}& \text{0}& \cdots & \text{0}\\ \text{0}& \text{0}& \text{1}& & \text{0}\\ ⋮& & & & ⋮\\ 0& \text{0}& \text{0}& & \text{1}\\ -{\text{a}}_0& -{\text{a}}_1& -{\text{a}}_2& & -{\text{a}}_{\text{n}-1}\end{array}\right]$

22. Let p(t) = a₀ + a₁t + a₂t² + t³, and let λ be a zero of p.

1. a. Write the companion matrix for p.
2. b. Explain why λ³ = −a₀ − a₁λ − a₂λ², and show that (1. λ, λ²) is an eigenvector of the companion matrix for p.