1.

1. (a) T

2. (b) T

3. (c) F

4. (d) F

5. (e) T

6. (f) T

7. (g) T

8. (h) F

9. (i) F

10. (j) T

2.

1. (a) $\left(\begin{array}{ll}0& 0\\ 0& 0\end{array}\right)$

2. (c) $\left(\begin{array}{cc}{\displaystyle \frac{7}{13}}& {\displaystyle \frac{7}{13}}\\ {\displaystyle \frac{6}{13}}& {\displaystyle \frac{6}{13}}\end{array}\right)$

3. (e) No limit exists.

4. (g) $\left(\begin{array}{rrr}-1& 0& -1\\ -4& 1& -2\\ 2& 0& 2\end{array}\right)$

5. (i) No limit exists.

6. For those who arrived during the week of June 1, after one week 25% of the patients recovered, 20% were ambulatory, 41% were bedridden, and 14% died. Eventually ${\displaystyle \frac{59}{90}}$ recover and ${\displaystyle \frac{31}{90}}$ die.

7. ${\displaystyle \frac{3}{7}}.$

8. Only the matrices in (a) and (b) are regular transition matrices.

9.

1. (a) $\left(\begin{array}{ccc}{\displaystyle \frac{1}{3}}& {\displaystyle \frac{1}{3}}& {\displaystyle \frac{1}{3}}\\ {\displaystyle \frac{1}{3}}& {\displaystyle \frac{1}{3}}& {\displaystyle \frac{1}{3}}\\ {\displaystyle \frac{1}{3}}& {\displaystyle \frac{1}{3}}& {\displaystyle \frac{1}{3}}\end{array}\right)$

2. (c) No limit exists.

3. (e) $\left(\begin{array}{ccc}0& 0& 0\\ {\displaystyle \frac{1}{2}}& 1& 0\\ {\displaystyle \frac{1}{2}}& 0& 1\end{array}\right)$

4. (g) $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& 1& 0\\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& 0& 1\end{array}\right)$

10.

1. (a) $\left(\begin{array}{l}0.225\\ 0.441\\ 0.334\end{array}\right)$ after two stages and $\left(\begin{array}{l}0.20\\ 0.60\\ 0.20\end{array}\right)$ eventually

2. (c) $\left(\begin{array}{l}0.372\\ 0.225\\ 0.403\end{array}\right)$ after two stages and $\left(\begin{array}{l}0.50\\ 0.20\\ 0.30\end{array}\right)$ eventually

3. (e) $\left(\begin{array}{l}0.329\\ 0.334\\ 0.337\end{array}\right)$ after two stages and $\left(\begin{array}{l}{\displaystyle \frac{1}{3}}\\ {\displaystyle \frac{1}{3}}\\ {\displaystyle \frac{1}{3}}\end{array}\right)$ eventually

12. ${\displaystyle \frac{9}{19}}$ new, ${\displaystyle \frac{6}{19}}$ once-used, and ${\displaystyle \frac{4}{19}}$ twice-used

13. In 1995, 24% will own large cars, 34% will own intermediate-sized cars, and 42% will own small cars; the corresponding eventual percentages are 10%, 30%, and 60%.

20. $e^O=I$ and $e^I=eI\text{.}$