91Ó°ÊÓ

In Exercises 9–16, find a basis for the eigenspace corresponding to each listed eigenvalue.

10. \(A = \left( {\begin{array}{*{20}{c}}{10}&{ - 9}\\4&{ - 2}\end{array}} \right)\), \(\lambda = 4\)

In Exercises 9–16, find a basis for the eigenspace corresponding to each listed eigenvalue.

11. \(A = \left( {\begin{array}{*{20}{c}}4&{ - 2}\\{ - 3}&9\end{array}} \right)\), \(\lambda = 10\)

In Exercises 9-18, construct the general solution of \(x' = Ax\) involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.

14. \(A = \left( {\begin{aligned}{ {20}{c}}{ - 2}&1\\{ - 8}&2\end{aligned}} \right)\)

Question: Is \(\lambda = 2\) an eigenvalue of \(\left( {\begin{array}{*{20}{c}}3&2\\3&8\end{array}} \right)\)? Why or why not?

Question: Without calculation, find one eigenvalue and two linearly independent eigenvectors of \(A = \left( {\begin{array}{*{20}{c}}5&5&5\\5&5&5\\5&5&5\end{array}} \right)\). Justify your answer.

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.

Question: Explain why a \({\bf{2}} \times {\bf{2}}\) matrix can have at most two distinct eigenvalues. Explain why an \(n \times n\) matrix can have at most n distinct eigenvalues.

Question: Show that if \(\lambda \) is an eigenvalue of A if and only if \(\lambda \) is an eigenvalue of \({A^T}\). (Find out how \(A - \lambda I\) and \({A^T} - \lambda I\) are related.)

Question: Consider an \(n \times n\) matrix A with the property that the row sums all equal the same number s. Show that s is an eigenvalue of A. (Hint: Find an eigenvector.)

Question: Is \(\lambda = - 2\) an eigenvalue of \(\left( {\begin{array}{*{20}{c}}7&3\\3&{ - 1}\end{array}} \right)\)? Why or why not?