(Solution) - a Show that the companion matrix C p of p x -(2025 Original AI-Free Solution)

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(a) Show that the companion matrix C(p) of p(x) = x2 + ax + b has characteristic polynomial ?2 + a? + b.
(b) Show that if A is an eigenvalue of the companion matrix C(p) in part (a), then

is an eigenvector of C(p) corresponding to ?.