Linear Algebra with Applications, 10th Edition

6.8

1. 1. 

   1. (a) ${\lambda }_1=4,{\lambda }_2=-1,{\mathbf{x}}_1=(3,2{)}^T$;

   2. (b) ${\lambda }_1=8,{\lambda }_2=3,{\mathbf{x}}_1=(1,2{)}^T$;

   3. (c) ${\lambda }_1=7,{\lambda }_2=2,{\lambda }_3=0,{\mathbf{x}}_1=(1,1,1{)}^T$

2. 2. 

   1. (a) ${\lambda }_1=3,{\lambda }_2=-1,{\mathbf{x}}_1=(3,1{)}^T$;

   2. (b) $\begin{array}{c}{\lambda }_1=2=2\text{ exp}(0),\hfill \\ {\lambda }_2=-2=2\text{ exp}(\pi ,i),{\mathbf{x}}_1=(1,1{)}^T\hfill \end{array}$;

   3. (c) $\begin{array}{c}{\lambda }_1=2=2\text{ exp}(0),\hfill \\ {\lambda }_2=-1+\sqrt{3}i=2\text{ exp}\left(\frac{2\pi i}{3}\right),\hfill \\ {\lambda }_3=-1-\sqrt{3}i=2\text{ exp}\left(\frac{4\pi i}{3}\right),\hfill \\ {\mathbf{x}}_1=(4,2,1{)}^T\hfill \end{array}$

3. 3. $x_1=70,000,x_2=56,000,x_3=44,000$

4. 4. $x_1=x_2=x_3$

5. 5. ${(I-A)}^{-1}=I+A+\cdots +A^{m-1}$

6. 6. 

   1. (a) ${(I-A)}^{-1}=\left[\begin{array}{ccc}\hfill 1& \hfill -1& 3\\ \hfill 0& \hfill 0& 1\\ 0& \hfill -1& 2\end{array}\right]$;

   2. (b) $\begin{array}{l}{A}^2=\left[\begin{array}{ccc}0& \hfill -2& 2\\ \hfill 0& \hfill 0& 0\\ 0& \hfill 0& 0\end{array}\right],\\ A^3=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\end{array}$

7. 7. (b) and (c) are reducible.

8. 8. Hint: See Theorem6.8.2 .

9. 9. 

   1. (b) Hint: Apply Perron’s theorem to B and C.

10. 11. Hint: Make use of the result from Exercise 10.

11. 12. Hint: Apply Perron’s theorem to $A^k\mathrm{.}$

12. 13. 

    1. (c) Hint: Show that if $c_1$ did equal 0, then ${\mathbf{y}}_j$ would approach the zero vector as $j\to \infty ,$ contradicting the result from part (b).

13. 15. 

    1. (d) $\begin{array}{ccc}\hfill \mathbf{w}& =\hfill & {(\frac{12}{29},\frac{12}{29},\frac{3}{29},\frac{2}{29})}^T\hfill \\ \approx \hfill & (\hfill \end{array}$