Chapter 5, Problem 21SE

Use mathematical induction to prove that for n ≥ 2,

$\begin{array}{ccc}\det ({\text{C}}_{\text{p}}-\lambda \text{I})& =& {(-1)}^{\text{n}}(a_0+{\text{a}}_1\lambda +\cdots +a_{\text{n}-1}+{\lambda }^{\text{n}})\\ & =& {(-1)}^{\text{n}}\text{p}(\lambda )\end{array}$

[Hint: Expanding by cofactors down the first column, show that det (C_p − λI) has the form (−λ) B + (−1)ⁿa₀, where B is a certain polynomial (by the induction assumption).]