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Q21SE

Expert-verified

Linear Algebra and its Applications

Found in: Page 267

Book edition 5th

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

Pages 483 pages

ISBN 978-03219822384

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Short Answer

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 \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

21. Use mathematical induction to prove that for \(n \ge {\bf{2}}\),\(\begin{aligned}{c}det\left( {{C_p} - \lambda I} \right) = {\left( { - {\bf{1}}} \right)^n}\left( {{a_{\bf{0}}} + {a_{\bf{1}}}\lambda + ... + {a_{n - {\bf{1}}}}{\lambda ^{n - {\bf{1}}}} + {\lambda ^n}} \right)\\ = {\left( { - {\bf{1}}} \right)^n}p\left( \lambda \right)\end{aligned}\)
(Hint: Expanding by cofactors down the first column, show that \(det\left( {{C_p} - \lambda I} \right)\) has the form \(\left( { - \lambda B} \right) + {\left( { - {\bf{1}}} \right)^n}{a_{\bf{0}}}\) where \(B\) is a certain polynomial (by the induction assumption).)

It is proved that for \(n = k + 1\), \(\det \left( {{C_p} - \lambda I} \right) = {\left( { - 1} \right)^{k + 1}}p\left( \lambda \right)\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

Apply mathematical induction to prove as shown below:

\(\begin{aligned}{c}\det \left( {{C_p} - \lambda I} \right) &= {\left( { - 1} \right)^n}\left( {{a_0} + {a_1}\lambda + ... + {a_{n - 1}}{\lambda ^{n - 1}} + {\lambda ^n}} \right)\\ &= {\left( { - 1} \right)^n}p\left( \lambda \right)\end{aligned}\)

Step 2: Apply mathematical induction to find whether it is true for \(n{\bf{ = 2}}\)

For \(n = 2\) the matrix is shown below:

\({C_p} = \left( {\begin{aligned}{*{20}{c}}0&1\\{ - {a_0}}&{ - {a_1}}\end{aligned}} \right)\)

Now find the determinant.

\(\begin{aligned}{c}\det \left( {{C_p} - \lambda I} \right) &= \left( { - \lambda } \right)\left( { - {a_1} - \lambda } \right) + {a_0}\\ &= {a_0} + {a_1}\lambda + {\lambda ^2}\\ &= {\left( { - 1} \right)^2}\left( {{a_0} + {a_1}\lambda + {\lambda ^2}} \right)\end{aligned}\)

Thus, the result is true for \(n = 2\).

Step 3: Apply mathematical induction to find whether it is true for \(n = k\)

Assume the result is true for \(n = k\).

\(\det \left( {{C_p} - \lambda I} \right) = {\left( { - 1} \right)^k}q\left( \lambda \right)\)

Now check the result for \(n = k + 1\).

Now find the determinant.

\(\begin{aligned}{c}\det \left( {{C_p} - \lambda I} \right) &= \det \left( {\begin{aligned}{*{20}{c}}{ - \lambda }&1&0&{...}&0\\0&{ - \lambda }&1&{}&0\\:&{}&{}&{}&:\\0&0&0&{}&1\\{ - {a_0}}&{ - {a_1}}&{ - {a_2}}&{...}&{ - {a_k} - \lambda }\end{aligned}} \right)\\ &= \left( { - \lambda } \right)\det + \left( {\begin{aligned}{*{20}{c}}{ - \lambda }&1&{...}&0\\:&:&:&:\\0&{...}&{...}&1\\{ - {a_1}}&{ - {a_2}}&{...}&{ - {a_k} - \lambda }\end{aligned}} \right) + 0... + {\left( { - 1} \right)^{k + 1}}{a_0}et\left( {\begin{aligned}{*{20}{c}}1&0&{...}&0\\0&1&:&:\\:&{...}&{...}&1\\0&0&{...}&1\end{aligned}} \right)\\ &= \left( { - \lambda } \right){\left( { - 1} \right)^k}q\left( \lambda \right) + {\left( { - 1} \right)^{k + 1}}{a_0}\\ &= \left( \lambda \right)\left( { - 1} \right){\left( { - 1} \right)^k}\left( {{a_1} + ... + {a_k}{\lambda ^{k - 1}} + {\lambda ^k}} \right) + {\left( { - 1} \right)^{k + 1}}{a_0}\\ &= {\left( { - 1} \right)^{k + 1}}p\left( \lambda \right)\end{aligned}\)

Thus, the result is true for \(n \ge 2\).

Hence, it is proved that for \(n = k + 1\), \(\det \left( {{C_p} - \lambda I} \right) = {\left( { - 1} \right)^{k + 1}}p\left( \lambda \right)\).

Most popular questions for Math Textbooks

Let \(A = \left( {\begin{aligned}{*{20}{c}}{.4}&{ - .3}\\{.4}&{1.2}\end{aligned}} \right)\). Explain why \({A^k}\) approaches \(\left( {\begin{aligned}{*{20}{c}}{ - .5}&{ - .75}\\1&{1.5}\end{aligned}} \right)\) as \(k \to \infty \).

Let\(\varepsilon = \left\{ {{{\bf{e}}_1},{{\bf{e}}_2},{{\bf{e}}_3}} \right\}\) be the standard basis for \({\mathbb{R}^3}\),\(B = \left\{ {{{\bf{b}}_1},{{\bf{b}}_2},{{\bf{b}}_3}} \right\}\) be a basis for a vector space \(V\) and\(T:{\mathbb{R}^3} \to V\) be a linear transformation with the property that

\(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_3} - {x_2}} \right){{\bf{b}}_1} - \left( {{x_1} - {x_3}} \right){{\bf{b}}_2} + \left( {{x_1} - {x_2}} \right){{\bf{b}}_3}\)

Compute\(T\left( {{{\bf{e}}_1}} \right)\), \(T\left( {{{\bf{e}}_2}} \right)\) and \(T\left( {{{\bf{e}}_3}} \right)\).
Compute \({\left( {T\left( {{{\bf{e}}_1}} \right)} \right)_B}\), \({\left( {T\left( {{{\bf{e}}_2}} \right)} \right)_B}\) and \({\left( {T\left( {{{\bf{e}}_3}} \right)} \right)_B}\).
Find the matrix for \(T\) relative to \(\varepsilon \), and\(B\).

Let\(B = \left\{ {{{\bf{b}}_1},{{\bf{b}}_2},{{\bf{b}}_3}} \right\}\) be a basis for a vector space\(V\). Find \(T\left( {3{{\bf{b}}_1} - 4{{\bf{b}}_2}} \right)\) when \(T\) isa linear transformation from \(V\) to \(V\) whose matrix relative to \(B\) is

\({\left( T \right)_B} = \left( {\begin{aligned}0&{}&{ - 6}&{}&1\\0&{}&5&{}&{ - 1}\\1&{}&{ - 2}&{}&7\end{aligned}} \right)\)

Define\(T:{{\rm P}_3} \to {\mathbb{R}^4}\)by\(T\left( {\bf{p}} \right) = \left( {\begin{aligned}{{\bf{p}}\left( { - 3} \right)}\\{{\bf{p}}\left( { - 1} \right)}\\{{\bf{p}}\left( 1 \right)}\\{{\bf{p}}\left( 3 \right)}\end{aligned}} \right)\).

Show that \(T\) is a linear transformation.
Find the matrix for \(T\) relative to the basis \(\left\{ {1,t,{t^2},{t^3}} \right\}\)for \({{\rm P}_3}\)and the standard basis for \({\mathbb{R}^4}\).

Question: Show that if \(A\) and \(B\) are similar, then \(\det A = \det B\).

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