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### Linear Algebra and its Applications

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).20. Let $$p\left( t \right){\bf{ = }}\left( {t{\bf{ - 2}}} \right)\left( {t{\bf{ - 3}}} \right)\left( {t{\bf{ - 4}}} \right){\bf{ = - 24 + 26}}t{\bf{ - 9}}{t^{\bf{2}}}{\bf{ + }}{t^{\bf{3}}}$$. Write the companion matrix for $$p\left( t \right)$$, and use techniques from chapter $${\bf{3}}$$ to find the characteristic polynomial.

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

See the step by step solution

## Step 1: Find the companion matrix

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).

Find the companion matrix for the given polynomial.

$$p\left( t \right) = - 24 + 26t - 9{t^2} + {t^3}$$

Therefore, on comparison we get,

{C_p} = \left( {\begin{aligned}{*{20}{c}}0&1&0\\0&0&1\\{24}&{ - 26}&9\end{aligned}} \right)

## Step 2: Find the characteristic polynomial

\begin{aligned}{c}\det \left( {{C_p} - \lambda I} \right) &= \det \left( {\begin{aligned}{*{20}{c}}{ - \lambda }&1&0\\0&{ - \lambda }&1\\{24}&{ - 26}&{9 - \lambda }\end{aligned}} \right)\\ &= \left( { - \lambda } \right)\left( {\left( { - \lambda } \right)\left( {9 - \lambda } \right) + 26} \right) + 24\\ &= \left( { - \lambda } \right)\left( { - 9\lambda + {\lambda ^2} + 26} \right) + 24\\ &= 9{\lambda ^2} - {\lambda ^3} - 26\lambda + 24\\ &= - \left( { - 24 + 26\lambda - 9{\lambda ^2} + {\lambda ^3}} \right)\\ &= - p\left( \lambda \right)\end{aligned}

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