StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q19SE

Expert-verifiedFound in: Page 267

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**Exercises 19–23 concern the polynomial \(p\left( t \right) = {a_{\bf{0}}} + {a_{\bf{1}}}t + ... + {a_{n - {\bf{1}}}}{t^{n - {\bf{1}}}} + {t^n}\) and \(n \times n\) matrix \({C_p}\) called the companion matrix of \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).**

** **

**19. Write the companion matrix \({C_p}\) for \(p\left( t \right) = {\bf{6}} - {\bf{5}}t + {t^{\bf{2}}}\), and then find the characteristic polynomial of \({C_p}\).**

** **

The characteristic polynomial of the matrix \({C_p}\)** **is** \(p\left( \lambda \right)\)**.

Consider the polynomial \(p\left( t \right) = {a_0} + {a_1}t + ... + {a_{n - 1}}{t^{n - 1}} + {t^n}\)**.**

The companion matrix of \(p\)** **is** \({C_p} = \left( {\begin{aligned}{*{20}{c}}0&1&0&{...}&0\\0&0&1&{}&0\\:&{}&{}&{}&:\\0&0&0&{}&1\\{ - {a_0}}&{ - {a_1}}&{ - {a_2}}&{...}&{ - {a_{n - 1}}}\end{aligned}} \right)\).**

Thus, we get,

\({C_p} = \left( {\begin{aligned}{*{20}{c}}0&1\\{ - 6}&5\end{aligned}} \right)\)

\(\begin{aligned}{c}\det \left( {{C_p} - \lambda I} \right) &= \det \left( {\begin{aligned}{*{20}{c}}{0 - \lambda }&1\\{ - 6}&{5 - \lambda }\end{aligned}} \right)\\ &= \left( { - \lambda } \right)\left( {5 - \lambda } \right) + 6\\ &= - 5\lambda + {\lambda ^2} + 6\\ &= 6 - 5\lambda + {\lambda ^2}\\ &= p\left( \lambda \right)\end{aligned}\)

Therefore, the characteristic polynomial of the matrix \({C_p}\)** **is** \(p\left( \lambda \right)\)**.

94% of StudySmarter users get better grades.

Sign up for free