Chapter 5, Problem 20SE

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

20. Let p(t) = (t − 2)(t − 3)(t − 4) = −24 + 26t − 9t² + t³ Write the companion matrix for p(t), and use techniques from Chapter 3 to find its characteristic polynomial.