Exercise 1.3.13. We have two sets of people, those who start the year residing in California and those who do not. 80% of those starting the year in California are still in California at the end of the year, while 20% of those have moved elsewhere. Of those residing outside California at the beginning of the year, 10% have moved to California by the end of the year, while 90% still reside elsewhere.

Let u be the number of people outside California and v be the number of people in California. If u and v at the end of the year equal u and v at the start, determine the steady-state ratio of u to v.

Answer: Assuming u is the number of people outside California at the start of the year, 0.9u is the number of those people still outside at the end of the year. Similarly, if v is the number of people in California at the start of the year, 0.2v is the number of those people who’ve moved outside at the end of the year. So the number of people outside California at the end of the year is

Similarly, 0.1u is the number of those people residing outside California who have moved there at the end of the year, and 0.8v is the number of people in California at the start of the year who are still there at the end of the year. So the number of people residing in California at the end of the year is

So in both cases we have

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.