Nice presentation!
Q1
1a) Good. Thanks for getting Maple to check the sums.
1b) Good k by k proof. Could you have formulated the proof by induction carefully? (Though it's clear that you get it.) Good use of the 3 by 3 case as an example.
1c) Excellent. You might want to note, as well, that the entries of D must be positive.
1d) Your exponent is impressive. I'm surprised Maple computes it so quickly. Correct guess. Your proof that DA=D is good, and relevant, but in the second half you fail, since you used the assumption about D that you haven't proved. Instead, take DA=D, prove D.A^n=D by induction, then conclude that DD=D by taking limits.
Q2
2a) I didn't know that Maple output could be pleasant to look at.
2b) Excellent.
2c) Fine. Could you give more explanation of how you got your expression for x_n? Could you recognize and sum a GEOMETRIC SERIES without getting Maple to tell you?
Q3 I think that this is a nice list, but I'm not sure that I agree with your division into parameters and constants. Do you have enough variables? Your model is changing over time, so probably time ought to be a variable, and then the number of people left in the city at time t ought to be another variable, don't you think?