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Found in: Page 267

### 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).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 $$p\left( \lambda \right)$$.

See the step by step solution

## Step 1: Compare the given companion matrix with the given polynomial.

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)

## Step 2: Find the characteristic polynomial

\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 $$p\left( \lambda \right)$$.

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