Solution Found!
Answer: Exercises 19–23 concern the polynomial and an n ×
Chapter , Problem 21E(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 \(n \times n\) matrix \(C_p\) called the companion matrix of p:
\(C_p=\left[\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & & 1 \\ -a_0 & -a_1 & -a_2 & \ldots & -a_{n-1} \end{array}\right]\)
Use mathematical induction to prove that for \(n \geq 2\),
\(\begin{aligned} \operatorname{det}\left(C_p-\lambda I\right) &=(-1)^n\left(a_0+a_1 \lambda+\cdots+a_{n-1} \lambda^{n-1}+\lambda^n\right) \\ &=(-1)^n p(\lambda) \end{aligned}\)
[Hint: Expanding by cofactors down the first column, show that \(\operatorname{det}\left(C_p-\lambda I\right)\) has the form \((-\lambda) B+(-1)^n a_0\), where B is a certain polynomial (by the induction assumption).]
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 \(n \times n\) matrix \(C_p\) called the companion matrix of p:
\(C_p=\left[\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & & 1 \\ -a_0 & -a_1 & -a_2 & \ldots & -a_{n-1} \end{array}\right]\)
Use mathematical induction to prove that for \(n \geq 2\),
\(\begin{aligned} \operatorname{det}\left(C_p-\lambda I\right) &=(-1)^n\left(a_0+a_1 \lambda+\cdots+a_{n-1} \lambda^{n-1}+\lambda^n\right) \\ &=(-1)^n p(\lambda) \end{aligned}\)
[Hint: Expanding by cofactors down the first column, show that \(\operatorname{det}\left(C_p-\lambda I\right)\) has the form \((-\lambda) B+(-1)^n a_0\), where B is a certain polynomial (by the induction assumption).]
ANSWER:Solution 21E