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Q6E

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Linear Algebra and its Applications
Found in: Page 267
Linear Algebra and its Applications

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

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Short Answer

Question: Is \(\left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\) an eigenvector of\(\left){\begin{array}{*{20}{c}}3&6&7\\3&3&7\\5&6&5\end{array}} \right)\)? If so, find the eigenvalue.

Yes, \(\left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\) is the eigenvector of the given matrix \(\left( {\begin{array}{*{20}{c}}3&6&7\\3&3&7\\5&6&5\end{array}} \right)\) , and the eigenvalue is \( - 2\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Definition of eigenvector

If there exists a non-zero vector \({\bf{x}}\) that satisfies \(A{\bf{x}} = \lambda {\bf{x}}\) for some scaler \(\lambda \), then \({\bf{x}}\) be the eigenvector of an \(n \times n\) matrix \(A\), and if \(A{\bf{x}} = \lambda {\bf{x}}\) exists, then scaler \(\lambda \) is the eigenvalue of the matrix.

Step 2: Determine whether \(\left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\) is the eigenvector of the given matrix

Denote the given matrix by \(A\) and the given vector by \({\bf{x}}\).

\(A = \left( {\begin{array}{*{20}{c}}3&6&7\\3&3&7\\5&6&5\end{array}} \right)\), \({\bf{x}} = \left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\)

According to the definition of eigenvalue, \({\bf{x}} = \left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\) is the eigenvector of the matrix \(A\), if \(A{\bf{x}} = \lambda {\bf{x}}\).

Find the product of \(A\), and \({\bf{x}}\).

\(\begin{array}{c}A{\bf{x}} = \left( {\begin{array}{*{20}{c}}3&6&7\\3&3&7\\5&6&5\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{3\left( 1 \right) + 6\left( { - 2} \right) + 7\left( 1 \right)}\\{3\left( 1 \right) + 3\left( { - 2} \right) + 7\left( 1 \right)}\\{5\left( 1 \right) + 6\left( { - 2} \right) + 5\left( 1 \right)}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{3 - 12 + 7}\\{3 - 6 + 7}\\{5 - 12 + 5}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{ - 2}\\4\\{ - 2}\end{array}} \right)\end{array}\)

The obtained matrix in the form of the given vector can be written as:

\(A{\bf{x}} = \left( { - 2} \right)\left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\)

So, the vector \(\left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\) is the eigenvector of the given matrix \(\left( {\begin{array}{*{20}{c}}3&6&7\\3&3&7\\5&6&5\end{array}} \right)\).

Step 3: Determine the eigenvalue 

As the given vector satisfies the condition \(A{\bf{x}} = \lambda {\bf{x}}\). Which implies that \(\lambda \) is the eigenvalue of the given matrix. So, \(\lambda = - 2\).

So, \( - 2\) is the eigenvalue of the given matrix \(\left( {\begin{array}{*{20}{c}}3&6&7\\3&3&7\\5&6&5\end{array}} \right)\).

Most popular questions for Math Textbooks

Suppose \({\bf{x}}\) is an eigenvector of \(A\) corresponding to an eigenvalue \(\lambda \).

a. Show that \(x\) is an eigenvector of \(5I - A\). What is the corresponding eigenvalue?

b. Show that \(x\) is an eigenvector of \(5I - 3A + {A^2}\). What is the corresponding eigenvalue?

Question: Let \(A = \left( {\begin{array}{*{20}{c}}{ - 6}&{28}&{21}\\4&{ - 15}&{ - 12}\\{ - 8}&a&{25}\end{array}} \right)\). For each value of \(a\) in the set \(\left\{ {32,31.9,31.8,32.1,32.2} \right\}\), compute the characteristic polynomial of \(A\) and the eigenvalues. In each case, create a graph of the characteristic polynomial \(p\left( t \right) = \det \left( {A - tI} \right)\) for \(0 \le t \le 3\). If possible, construct all graphs on one coordinate system. Describe how the graphs reveal the changes in the eigenvalues of \(a\) changes.

Question: Is \(\left( {\begin{array}{*{20}{c}}1\\4\end{array}} \right)\) an eigenvalue of \(\left( {\begin{array}{*{20}{c}}{ - 3}&1\\{ - 3}&8\end{array}} \right)\)? If so, find the eigenvalue.

Question: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.

6. \(\left[ {\begin{array}{*{20}{c}}3&- 4\\4&8\end{array}} \right]\)

Define \(T:{{\rm P}_2} \to {\mathbb{R}^3}\) by \(T\left( {\bf{p}} \right) = \left( {\begin{aligned}{{\bf{p}}\left( { - 1} \right)}\\{{\bf{p}}\left( 0 \right)}\\{{\bf{p}}\left( 1 \right)}\end{aligned}} \right)\).

  1. Find the image under\(T\)of\({\bf{p}}\left( t \right) = 5 + 3t\).
  2. Show that \(T\) is a linear transformation.
  3. Find the matrix for \(T\) relative to the basis \(\left\{ {1,t,{t^2}} \right\}\)for \({{\rm P}_2}\)and the standard basis for \({\mathbb{R}^3}\).
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