I've done that question before.
Basically if the family goes to P this year, they cannot go to P next year and so on.
Start by eliminating transition matrices that fail to depict this situation. If you look at A, you will see that for any given year, the family will always return to the SAME exact location next year, 100% of the time. Rule out A.
For B, if the family go to R one year, they will 100% return to R next year. Rule out B.
For C, if a family goes to Q one year, they have a 40% chance of returning next year. Rule out C.
For D, the transition columns do not add to 1, so therefore it is an invalid transition matrix. Rule out D.
For E, there is 0% chance that a family can spend two years in the exact same location. This is true for all destinations. Therefore E is correct.