(Solution) - a Use mathematical induction to prove that for n -(2025 Original AI-Free Solution)

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(a) Use mathematical induction to prove that, for n ? 2, the companion matrix C(p) of p (x) = xn + an-1xn-1 + ··· + a1x + a0 has characteristic polynomial (-1) np (A).
(b) Show that if A is an eigenvalue of the companion matrix C(p) in Equation (4), then an eigenvector corresponding to ? is given by

If p(x) = xn + an-1xn-1 + ··· + a1x + a0 and A is a square matrix, we can define a square matrix p(A) by
p(A) = An + an-1An-i + ··· + a1A + a0I