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Solved: Exercises 19–23 concern the polynomial and an n ×
Chapter , Problem 20E(choose chapter or problem)
Exercises 19–23 concern the polynomial
\(p(t)=a_{0}+a_{1} t+\cdots+a_{n-1} t^{n-1}+t^{n}\)
and an n X n matrix \(C_p\) called the companion matrix of p:
\(C_{p}=\left[\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & & 1 \\ -a_{0} & -a_{1} & -a_{2} & \cdots & -a_{n-1} \end{array}\right]\)
Let \(p(t)=(t-2)(t-3)(t-4)=-24+26 t-9 t^{2}+t^{3}\). Write the companion matrix for p(t), and use techniques from Chapter 3 to find its characteristic polynomial.
Questions & Answers
QUESTION:
Exercises 19–23 concern the polynomial
\(p(t)=a_{0}+a_{1} t+\cdots+a_{n-1} t^{n-1}+t^{n}\)
and an n X n matrix \(C_p\) called the companion matrix of p:
\(C_{p}=\left[\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & & 1 \\ -a_{0} & -a_{1} & -a_{2} & \cdots & -a_{n-1} \end{array}\right]\)
Let \(p(t)=(t-2)(t-3)(t-4)=-24+26 t-9 t^{2}+t^{3}\). Write the companion matrix for p(t), and use techniques from Chapter 3 to find its characteristic polynomial.
ANSWER:Solution 20E