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Chapter 5: Q21E (page 267)

Question: In Exercises 21 and 22, A is an\(n \times n\)matrix. Mark each statement True or False. Justify each answer.

  1. If\(A{\bf{x}} = \lambda {\bf{x}}\)for some vector x, then\(\lambda \)is an eignvalue of A.
  2. A matrix A is not invertible if and only if 0 is an eigenvalue of A.
  3. A number of c is an eigenvalue of A if and only if the equation\(\left( {A - cI} \right){\bf{x}} = {\bf{0}}\)has a nontrivial solution.
  4. Finding an eigenvector of A may be difficult, but checking whether a given vector is in face an eigenvector is easy.
  5. To find the eigenvalues of A, reduce A to echelon form.

Short Answer

Expert verified

a. The given statement is False.

b. The given statement is True.

c. The given statement is True

d. The given statement is True.

e. The given statement is False.

Step by step solution

01

Find an answer for part (a)

The equation \(A{\bf{x}} = \lambda {\bf{x}}\) essentially has a nontrivial solution, then only \(\lambda \) will be the eigenvalue of A.

Thus, statement (a) is false.

02

Find an answer for part (b)

If A is invertible, then 0 is an eigenvalue of A.

Thus, statement (b) is true.

03

Find an answer for part (c)

If a scalar c is the eigenvalue of A, then;

\(\begin{array}{c}A{\bf{x}} = c{\bf{x}}\\\left( {A - c} \right){\bf{x}} = 0\end{array}\)

04

Find an answer for part (d)

To find an eigenvector of a matrix, first, it needs to determine the eigenvalue and then the eigenvector.

On the other hand, to check the eigenvector, we need to solve the equation \(A{\bf{x}} = \lambda {\bf{x}}\), where x is an eigenvector.

Thus, statement (d) is true.

05

Find the answer for part (e)

The characteristic equation of the matrix A is:

\(\det \left( {A - \lambda I} \right) = 0\)

The eigenvalues cannot be created by reducing in echelon form. The echelon form is used to determine the eigenvectors.

Thus, the statement (e) is false.

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Most popular questions from this chapter

Question: For the matrices in Exercises 15-17, list the eigenvalues, repeated according to their multiplicities.

16. \(\left[ {\begin{array}{*{20}{c}}5&0&0&0\\8&- 4&0&0\\0&7&1&0\\1&{ - 5}&2&1\end{array}} \right]\)

M] In Exercises 19 and 20, find (a) the largest eigenvalue and (b) the eigenvalue closest to zero. In each case, set \[{{\bf{x}}_{\bf{0}}}{\bf{ = }}\left( {{\bf{1,0,0,0}}} \right)\] and carry out approximations until the approximating sequence seems accurate to four decimal places. Include the approximate eigenvector.

19.\[A{\bf{=}}\left[{\begin{array}{*{20}{c}}{{\bf{10}}}&{\bf{7}}&{\bf{8}}&{\bf{7}}\\{\bf{7}}&{\bf{5}}&{\bf{6}}&{\bf{5}}\\{\bf{8}}&{\bf{6}}&{{\bf{10}}}&{\bf{9}}\\{\bf{7}}&{\bf{5}}&{\bf{9}}&{{\bf{10}}}\end{array}} \right]\]

Let \(A{\bf{ = }}\left( {\begin{aligned}{*{20}{c}}{{a_{{\bf{11}}}}}&{{a_{{\bf{12}}}}}\\{{a_{{\bf{21}}}}}&{{a_{{\bf{22}}}}}\end{aligned}} \right)\). Recall from Exercise \({\bf{25}}\) in Section \({\bf{5}}{\bf{.4}}\) that \({\rm{tr}}\;A\) (the trace of \(A\)) is the sum of the diagonal entries in \(A\). Show that the characteristic polynomial of \(A\) is \({\lambda ^2} - \left( {{\rm{tr}}A} \right)\lambda + \det A\). Then show that the eigenvalues of a \({\bf{2 \times 2}}\) matrix \(A\) are both real if and only if \(\det A \le {\left( {\frac{{{\rm{tr}}A}}{2}} \right)^2}\).

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

Use mathematical induction to show that if \(\lambda \) is an eigenvalue of an \(n \times n\) matrix \(A\), with a corresponding eigenvector, then, for each positive integer \(m\), \({\lambda ^m}\)is an eigenvalue of \({A^m}\), with \({\rm{x}}\) a corresponding eigenvector.
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