Section 6.3

1. $P = [\begin{array}{r} 1 & 2 \\ 1 & 1 \end{array}], D = [\begin{array}{r} 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 63 & - 62 \\ 31 & - 30 \end{array}]$ $P = [\begin{array}{r} 1 & 2 \\ 1 & 1 \end{array}], D = [\begin{array}{r} 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 63 & - 62 \\ 31 & - 30 \end{array}]$
2. $P = [\begin{array}{r} 1 & 2 \\ 1 & 1 \end{array}], D = [\begin{array}{r} - 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 65 & - 66 \\ 33 & - 34 \end{array}]$ $P = [\begin{array}{r} 1 & 2 \\ 1 & 1 \end{array}], D = [\begin{array}{r} - 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 65 & - 66 \\ 33 & - 34 \end{array}]$
3. $P = [\begin{array}{r} 1 & 3 \\ 1 & 2 \end{array}], D = [\begin{array}{r} 0 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 96 & - 96 \\ 64 & - 64 \end{array}]$ $P = [\begin{array}{r} 1 & 3 \\ 1 & 2 \end{array}], D = [\begin{array}{r} 0 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 96 & - 96 \\ 64 & - 64 \end{array}]$
4. $P = [\begin{array}{r} 1 & 3 \\ 1 & 2 \end{array}], D = [\begin{array}{r} 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 94 & - 93 \\ 62 & - 61 \end{array}]$ $P = [\begin{array}{r} 1 & 3 \\ 1 & 2 \end{array}], D = [\begin{array}{r} 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 94 & - 93 \\ 62 & - 61 \end{array}]$
5. $P = [\begin{array}{r} 1 & 4 \\ 1 & 3 \end{array}], D = [\begin{array}{r} 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 125 & - 124 \\ 93 & - 92 \end{array}]$ $P = [\begin{array}{r} 1 & 4 \\ 1 & 3 \end{array}], D = [\begin{array}{r} 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 125 & - 124 \\ 93 & - 92 \end{array}]$
6. $P = [\begin{array}{r} 2 & 5 \\ 1 & 1 \end{array}], D = [\begin{array}{r} 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 156 & - 310 \\ 62 & - 123 \end{array}]$ $P = [\begin{array}{r} 2 & 5 \\ 1 & 1 \end{array}], D = [\begin{array}{r} 1 & 0 \\ 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 156 & - 310 \\ 62 & - 123 \end{array}]$
7. $P = [\begin{array}{r} 1 & 0 & 3 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}], D = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 1 & 93 & 0 \\ 0 & 32 & 0 \\ 0 & 0 & 32 \end{array}]$ $P = [\begin{array}{r} 1 & 0 & 3 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}], D = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 1 & 93 & 0 \\ 0 & 32 & 0 \\ 0 & 0 & 32 \end{array}]$
8. $P = [\begin{array}{r} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 2 & 0 & 1 \end{array}], D = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 1 & - 62 & 31 \\ 0 & 1 & 0 \\ 0 & - 62 & 32 \end{array}]$ $P = [\begin{array}{r} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 2 & 0 & 1 \end{array}], D = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 1 & - 62 & 31 \\ 0 & 1 & 0 \\ 0 & - 62 & 32 \end{array}]$
9. $P = [\begin{array}{r} 1 & 1 & - 3 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}], D = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 1 & - 93 & 31 \\ 0 & 32 & 0 \\ 0 & 0 & 32 \end{array}]$ $P = [\begin{array}{r} 1 & 1 & - 3 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}], D = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 1 & - 93 & 31 \\ 0 & 32 & 0 \\ 0 & 0 & 32 \end{array}]$
10. $P = [\begin{array}{r} 1 & - 1 & 3 \\ 1 & 0 & 2 \\ 0 & 2 & 0 \end{array}], D = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 94 & - 93 & 31 \\ 62 & - 61 & 31 \\ 0 & 0 & 32 \end{array}]$ $P = [\begin{array}{r} 1 & - 1 & 3 \\ 1 & 0 & 2 \\ 0 & 2 & 0 \end{array}], D = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}], A^{5} = [\begin{array}{r} 94 & - 93 & 31 \\ 62 & - 61 & 31 \\ 0 & 0 & 32 \end{array}]$
11. $P = [\begin{array}{r} 0 & 0 & - 1 \\ - 1 & - 2 & - 42 \\ 3 & 5 & 87 \end{array}], D = [\begin{array}{r} - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}], A^{10} = [\begin{array}{r} 1 & 0 & 0 \\ 78 & - 5 & - 2 \\ - 195 & 15 & 6 \end{array}]$ $P = [\begin{array}{r} 0 & 0 & - 1 \\ - 1 & - 2 & - 42 \\ 3 & 5 & 87 \end{array}], D = [\begin{array}{r} - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}], A^{10} = [\begin{array}{r} 1 & 0 & 0 \\ 78 & - 5 & - 2 \\ - 195 & 15 & 6 \end{array}]$
12. $P = [\begin{array}{r} 1 & 1 & 3 \\ 2 & 0 & 5 \\ 0 & 5 & 0 \end{array}], D = [\begin{array}{r} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}], A^{10} = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$ $P = [\begin{array}{r} 1 & 1 & 3 \\ 2 & 0 & 5 \\ 0 & 5 & 0 \end{array}], D = [\begin{array}{r} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}], A^{10} = [\begin{array}{r} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$
13. $P = [\begin{array}{r} - 1 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 0 & 1 \end{array}], D = [\begin{array}{r} - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}], A^{10} = [\begin{array}{r} 3 & - 3 & 1 \\ 2 & - 2 & 1 \\ 0 & 0 & 1 \end{array}]$ $P = [\begin{array}{r} - 1 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 0 & 1 \end{array}], D = [\begin{array}{r} - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}], A^{10} = [\begin{array}{r} 3 & - 3 & 1 \\ 2 & - 2 & 1 \\ 0 & 0 & 1 \end{array}]$
14. $P = [\begin{array}{r} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 2 & 0 & 1 \end{array}], D = [\begin{array}{r} - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}], A^{10} = [\begin{array}{r} 3 & - 3 & 1 \\ 2 & - 2 & 1 \\ 0 & 0 & 1 \end{array}]$ $P = [\begin{array}{r} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 2 & 0 & 1 \end{array}], D = [\begin{array}{r} - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}], A^{10} = [\begin{array}{r} 3 & - 3 & 1 \\ 2 & - 2 & 1 \\ 0 & 0 & 1 \end{array}]$
15. $A^{2} - 3 A + 2 I = 0, A^{3} = [\begin{array}{l} 29 & - 28 \\ 21 & - 20 \end{array}], A^{4} = [\begin{array}{l} 61 & - 60 \\ 45 & - 44 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{l} - 2 & 4 \\ - 3 & 5 \end{array}]$ $A^{2} - 3 A + 2 I = 0, A^{3} = [\begin{array}{l} 29 & - 28 \\ 21 & - 20 \end{array}], A^{4} = [\begin{array}{l} 61 & - 60 \\ 45 & - 44 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{l} - 2 & 4 \\ - 3 & 5 \end{array}]$
16. $A^{2} - 3 A + 2 I = 0, A^{3} = [\begin{array}{l} 36 & - 70 \\ 14 & - 27 \end{array}], A^{4} = [\begin{array}{l} 76 & - 150 \\ 30 & - 59 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{l} - 3 & 10 \\ - 2 & 6 \end{array}]$ $A^{2} - 3 A + 2 I = 0, A^{3} = [\begin{array}{l} 36 & - 70 \\ 14 & - 27 \end{array}], A^{4} = [\begin{array}{l} 76 & - 150 \\ 30 & - 59 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{l} - 3 & 10 \\ - 2 & 6 \end{array}]$
17. $- A^{3} + 5 A^{2} - 8 A + 4 I = 0, A^{3} = [\begin{array}{r} 1 & 21 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{array}], A^{4} = [\begin{array}{r} 1 & 45 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 16 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{r} 2 & - 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$ $- A^{3} + 5 A^{2} - 8 A + 4 I = 0, A^{3} = [\begin{array}{r} 1 & 21 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{array}], A^{4} = [\begin{array}{r} 1 & 45 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 16 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{r} 2 & - 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$
18. $- A^{3} + 4 A^{2} - 5 A + 2 I = 0, A^{3} = [\begin{array}{r} 1 & - 14 & 7 \\ 0 & 1 & 0 \\ 0 & - 14 & 8 \end{array}], A^{4} = [\begin{array}{r} 1 & - 30 & 15 \\ 0 & 1 & 0 \\ 0 & - 30 & 16 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{r} 2 & 2 & - 1 \\ 0 & 2 & 0 \\ 0 & 2 & 1 \end{array}]$ $- A^{3} + 4 A^{2} - 5 A + 2 I = 0, A^{3} = [\begin{array}{r} 1 & - 14 & 7 \\ 0 & 1 & 0 \\ 0 & - 14 & 8 \end{array}], A^{4} = [\begin{array}{r} 1 & - 30 & 15 \\ 0 & 1 & 0 \\ 0 & - 30 & 16 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{r} 2 & 2 & - 1 \\ 0 & 2 & 0 \\ 0 & 2 & 1 \end{array}]$
19. $- A^{3} + 5 A^{2} - 8 A + 4 I = 0, A^{3} = [\begin{array}{r} 1 & - 21 & 7 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{array}], A^{4} = [\begin{array}{r} 1 & - 45 & 15 \\ 0 & 16 & 0 \\ 0 & 0 & 16 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{r} 2 & 3 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$ $- A^{3} + 5 A^{2} - 8 A + 4 I = 0, A^{3} = [\begin{array}{r} 1 & - 21 & 7 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{array}], A^{4} = [\begin{array}{r} 1 & - 45 & 15 \\ 0 & 16 & 0 \\ 0 & 0 & 16 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{r} 2 & 3 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$
20. $- A^{3} + 5 A^{2} - 8 A + 4 I = 0, A^{3} = [\begin{array}{r} 22 & - 21 & 7 \\ 14 & - 13 & 7 \\ 0 & 0 & 8 \end{array}], A^{4} = [\begin{array}{r} 46 & - 45 & 15 \\ 30 & - 29 & 15 \\ 0 & 0 & 16 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{r} - 1 & 3 & - 1 \\ - 2 & 4 & - 1 \\ 0 & 0 & 1 \end{array}]$ $- A^{3} + 5 A^{2} - 8 A + 4 I = 0, A^{3} = [\begin{array}{r} 22 & - 21 & 7 \\ 14 & - 13 & 7 \\ 0 & 0 & 8 \end{array}], A^{4} = [\begin{array}{r} 46 & - 45 & 15 \\ 30 & - 29 & 15 \\ 0 & 0 & 16 \end{array}], A^{- 1} = \frac{1}{2} [\begin{array}{r} - 1 & 3 & - 1 \\ - 2 & 4 & - 1 \\ 0 & 0 & 1 \end{array}]$
21. $- A^{3} + A = 0$ $- A^{3} + A = 0$ , $A^{3} = A = A = [\begin{array}{r} 1 & 0 & 0 \\ 6 & 5 & 2 \\ 21 & - 15 & - 6 \end{array}], A^{4} = A^{2} = [\begin{array}{r} 1 & 0 & 0 \\ 78 & - 5 & - 2 \\ - 195 & 15 & 6 \end{array}] .$ $A^{3} = A = A = [\begin{array}{r} 1 & 0 & 0 \\ 6 & 5 & 2 \\ 21 & - 15 & - 6 \end{array}], A^{4} = A^{2} = [\begin{array}{r} 1 & 0 & 0 \\ 78 & - 5 & - 2 \\ - 195 & 15 & 6 \end{array}] .$ Because $λ = 0$ $λ = 0$ is an eigenvalue, A is singular and $A^{- 1}$ $A^{- 1}$ does not exist.
22. $- A^{3} + A^{2} + A - I = 0, A^{3} = A = [\begin{array}{l} 11 & - 6 & - 2 \\ 20 & - 11 & - 4 \\ 0 & 0 & 1 \end{array}] = A, A^{4} = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] = I, A^{- 1} = A$ $- A^{3} + A^{2} + A - I = 0, A^{3} = A = [\begin{array}{l} 11 & - 6 & - 2 \\ 20 & - 11 & - 4 \\ 0 & 0 & 1 \end{array}] = A, A^{4} = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] = I, A^{- 1} = A$
23. $- A^{3} + A = 0$ $- A^{3} + A = 0$ , $A^{3} = A = [\begin{array}{r} 1 & - 1 & 1 \\ 2 & - 2 & 1 \\ 4 & - 4 & 1 \end{array}], A^{4} = A^{2} = [\begin{array}{r} 3 & - 3 & 1 \\ 2 & - 2 & 1 \\ 0 & 0 & 1 \end{array}] .$ $A^{3} = A = [\begin{array}{r} 1 & - 1 & 1 \\ 2 & - 2 & 1 \\ 4 & - 4 & 1 \end{array}], A^{4} = A^{2} = [\begin{array}{r} 3 & - 3 & 1 \\ 2 & - 2 & 1 \\ 0 & 0 & 1 \end{array}] .$ Because $λ = 0$ $λ = 0$ is an eigenvalue, A is singular and $A^{- 1}$ $A^{- 1}$ does not exist.
24. $- A^{3} + A = 0, A^{3} = A = [\begin{array}{r} 5 & - 5 & - 3 \\ 2 & - 2 & - 1 \\ 4 & - 4 & - 3 \end{array}], A^{4} = A^{2} = [\begin{array}{r} 3 & - 3 & - 1 \\ 2 & - 2 & - 1 \\ 0 & 0 & 1 \end{array}] .$ $- A^{3} + A = 0, A^{3} = A = [\begin{array}{r} 5 & - 5 & - 3 \\ 2 & - 2 & - 1 \\ 4 & - 4 & - 3 \end{array}], A^{4} = A^{2} = [\begin{array}{r} 3 & - 3 & - 1 \\ 2 & - 2 & - 1 \\ 0 & 0 & 1 \end{array}] .$ Because $λ = 0$ $λ = 0$ is an eigenvalue, A is singular and $A^{- 1}$ $A^{- 1}$ does not exist.
25. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 1 & - 1 \\ 1 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 4 / 5 \end{array}]}^{k} \frac{1}{2} [\begin{array}{r} 1 & 1 \\ - 1 & 1 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 1 / 2 \\ 1 / 2 \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 1 & - 1 \\ 1 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 4 / 5 \end{array}]}^{k} \frac{1}{2} [\begin{array}{r} 1 & 1 \\ - 1 & 1 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 1 / 2 \\ 1 / 2 \end{array}]$ as $k \to \infty$ $k \to \infty$ . The long-term distribution of population is 50% city, 50% suburban.
26. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 1 & - 1 \\ 3 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 4 / 5 \end{array}]}^{k} \frac{1}{4} [\begin{array}{r} 1 & 1 \\ - 3 & 1 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 1 / 4 \\ 3 / 4 \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 1 & - 1 \\ 3 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 4 / 5 \end{array}]}^{k} \frac{1}{4} [\begin{array}{r} 1 & 1 \\ - 3 & 1 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 1 / 4 \\ 3 / 4 \end{array}]$ as $k \to \infty$ $k \to \infty$ . The long-term distribution of population is 25% city, 75% suburban.
27. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 3 & - 1 \\ 5 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 3 / 5 \end{array}]}^{k} \frac{1}{8} [\begin{array}{r} 1 & 1 \\ - 5 & 3 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 3 / 8 \\ 5 / 8 \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 3 & - 1 \\ 5 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 3 / 5 \end{array}]}^{k} \frac{1}{8} [\begin{array}{r} 1 & 1 \\ - 5 & 3 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 3 / 8 \\ 5 / 8 \end{array}]$ as $k \to \infty$ $k \to \infty$ . The long-term distribution of population is 3/8 city, 5/8 suburban.
28. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 1 & - 1 \\ 2 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 7 / 10 \end{array}]}^{k} \frac{1}{3} [\begin{array}{r} 1 & 1 \\ - 2 & 1 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 1 / 3 \\ 2 / 3 \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 1 & - 1 \\ 2 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 7 / 10 \end{array}]}^{k} \frac{1}{3} [\begin{array}{r} 1 & 1 \\ - 2 & 1 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 1 / 3 \\ 2 / 3 \end{array}]$ as $k \to \infty$ $k \to \infty$ . The long-term distribution of population is 1/3 city, 2/3 suburban.
29. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 1 & - 1 \\ 2 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 17 / 20 \end{array}]}^{k} \frac{1}{3} [\begin{array}{r} 1 & 1 \\ - 2 & 1 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 1 / 3 \\ 2 / 3 \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 1 & - 1 \\ 2 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 17 / 20 \end{array}]}^{k} \frac{1}{3} [\begin{array}{r} 1 & 1 \\ - 2 & 1 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 1 / 3 \\ 2 / 3 \end{array}]$ as $k \to \infty$ $k \to \infty$ . The long-term distribution of population is 1/3 city, 2/3 suburban.
30. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 3 & - 1 \\ 4 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 13 / 20 \end{array}]}^{k} \frac{1}{7} [\begin{array}{r} 1 & 1 \\ - 4 & 3 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 3 / 7 \\ 4 / 7 \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 3 & - 1 \\ 4 & 1 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 13 / 20 \end{array}]}^{k} \frac{1}{7} [\begin{array}{r} 1 & 1 \\ - 4 & 3 \end{array}] x_{0} \to = (C_{0} + S_{0}) [\begin{array}{r} 3 / 7 \\ 4 / 7 \end{array}]$ as $k \to \infty$ $k \to \infty$ . The long-term distribution of population is 3/7 city, 4/7 suburban.
31. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 5 & 5 \\ 4 & 2 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 4 / 5 \end{array}]}^{k} \frac{1}{10} [\begin{array}{r} - 2 & 5 \\ 4 & - 5 \end{array}] x_{0} \to = [\begin{array}{r} \begin{array}{c} 2.5 R_{0} - F_{0} \\ 2 R_{0} - 0.8 F_{0} \end{array} \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 5 & 5 \\ 4 & 2 \end{array}] {[\begin{array}{c} 1 & 0 \\ 0 & 4 / 5 \end{array}]}^{k} \frac{1}{10} [\begin{array}{r} - 2 & 5 \\ 4 & - 5 \end{array}] x_{0} \to = [\begin{array}{r} \begin{array}{c} 2.5 R_{0} - F_{0} \\ 2 R_{0} - 0.8 F_{0} \end{array} \end{array}]$ as $k \to \infty$ $k \to \infty$ . The fox-rabbit population approaches a stable situation with $2.5 R_{0} - F_{0}$ $2.5 R_{0} - F_{0}$ foxes and $2 R_{0} - 0.8 F_{0}$ $2 R_{0} - 0.8 F_{0}$ rabbits.
32. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 10 & 2 \\ 7 & 1 \end{array}] {[\begin{array}{c} 19 / 20 & 0 \\ 0 & 17 / 20 \end{array}]}^{k} \frac{1}{4} [\begin{array}{r} - 1 & 2 \\ 7 & - 10 \end{array}] x_{0} \to = [\begin{array}{r} \begin{array}{r} 0 \\ 0 \end{array} \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 10 & 2 \\ 7 & 1 \end{array}] {[\begin{array}{c} 19 / 20 & 0 \\ 0 & 17 / 20 \end{array}]}^{k} \frac{1}{4} [\begin{array}{r} - 1 & 2 \\ 7 & - 10 \end{array}] x_{0} \to = [\begin{array}{r} \begin{array}{r} 0 \\ 0 \end{array} \end{array}]$ as $k \to \infty$ $k \to \infty$ . The fox and rabbit population both die out.
33. $x_{k} = A^{k} x_{0} = [\begin{array}{r} 10 & 10 \\ 9 & 3 \end{array}] {[\begin{array}{c} 21 / 20 & 0 \\ 0 & 3 / 4 \end{array}]}^{k} \frac{1}{60} [\begin{array}{r} - 3 & 10 \\ 9 & - 10 \end{array}] x_{0} \approx \frac{1}{60} {(1.05)}^{k} (10 R_{0} - 3 F_{0}) [\begin{array}{r} 10 \\ 9 \end{array}]$ $x_{k} = A^{k} x_{0} = [\begin{array}{r} 10 & 10 \\ 9 & 3 \end{array}] {[\begin{array}{c} 21 / 20 & 0 \\ 0 & 3 / 4 \end{array}]}^{k} \frac{1}{60} [\begin{array}{r} - 3 & 10 \\ 9 & - 10 \end{array}] x_{0} \approx \frac{1}{60} {(1.05)}^{k} (10 R_{0} - 3 F_{0}) [\begin{array}{r} 10 \\ 9 \end{array}]$ as when k is sufficiently large. The fox and rabbit populations are both increasing at 5% per year, with 10 foxes for each 9 rabbits.
34. $A = P D P^{- 1} = [\begin{array}{r} 41 & - 30 \\ 56 & - 41 \end{array}] .$ $A = P D P^{- 1} = [\begin{array}{r} 41 & - 30 \\ 56 & - 41 \end{array}] .$ If n is even, then $D^{n} = I$ $D^{n} = I$ so $A^{n} = P D^{n} P^{- 1} = P I P^{- 1} = I$ $A^{n} = P D^{n} P^{- 1} = P I P^{- 1} = I$ . If n is odd, then $A^{n} = A^{n - 1} A = I A = A$ $A^{n} = A^{n - 1} A = I A = A$ . Thus $A^{99} = A$ $A^{99} = A$ and $A^{100} = I$ $A^{100} = I$ .
35. $λ = \pm 1$ $λ = \pm 1$ implies that $D^{n} = I$ $D^{n} = I$ if n is even, in which case $A^{n} = P D^{n} P^{- 1} = I$ $A^{n} = P D^{n} P^{- 1} = I$ .
36. $A^{2} = I$ $A^{2} = I$ , so $A^{3} = A^{2} A = I A = A$ $A^{3} = A^{2} A = I A = A$ , $A^{4} = A^{3} A = A^{2} = I$ $A^{4} = A^{3} A = A^{2} = I$ , and so forth.
37. $A^{2} = - I$ $A^{2} = - I$ , so $A^{3} = A^{2} A = - I A = - A$ $A^{3} = A^{2} A = - I A = - A$ , $A^{4} = A^{3} A = - A^{2} = I$ $A^{4} = A^{3} A = - A^{2} = I$ , and so forth.
38. If $B = [\begin{array}{r} 0 & 1 \\ 0 & 0 \end{array}]$ $B = [\begin{array}{r} 0 & 1 \\ 0 & 0 \end{array}]$ so $B^{2} = 0$ $B^{2} = 0$ , and it follows that $A^{n} = I + n B = [\begin{array}{l} 1 & n \\ 0 & 1 \end{array}] .$ $A^{n} = I + n B = [\begin{array}{l} 1 & n \\ 0 & 1 \end{array}] .$

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Table of Contents for Section 6.3

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Table of Contents for
Section 6.3