Just number 6!!!

There are no rural areas in the country of Sneezus-only suburbs and cities. Each year 5% of the city population moves to the suburbs, and 95% stay in the city. And each year 3% of the suburban population moves to the city, while the other 97% stay in the suburbs. Suppose in the year 2000, we had 6000 people living in the city, and 4000 living in the suburbs. For each nonnega- tive integer k, let the vt k be the vector in R2 where the first coordinate is the number of people living in the city and the second coordinate is the number of people living in the suburbs (where k is the number of years after the year 2000) So (6000, 4000). In the year 2001, based on the above information, we w have (95)(6000) + (03) (4000) = 5820 people living in the city, and (.05)(6000) + (.97)(4000) = 4180 people living in the suburbs. Thus z- (5820, 4180). This situation can be expressed with linear algebra. Let T denote the following transition matrix: 95 03 05 .97 (.95)(6000)+ (.03)(4000) (.05)(6000)+ (.97)(4000)4180 5820 95 .03 6000 05 .974000 Thus Tzo = Thus z| = Tzo. Use MATLAB to answer the following questions 1, what is z2? (Remember, each year 5% of the city population moves to the suburbs, and 95% stay in the city. And each year 3% of the suburban population moves to the city, while the other 97% stay in the suburbs. ) 2. Can you give a formula for r2 in terms of the matrix T and o? How about for r3? 3. About how many people will be living in the city in the year 2005? How many people will be living in the suburbs in the year 2005? 4. Suppose in a given year there are 3750 people living in the city and 6250 people living in the suburbs. How many people l be living in the city and how many in the suburbs the following year? two years later? 3750 6250 5. Let s be the vector s- Why would you think s is called a "steady-state vector"? (If you have no idea, you may have done something wrong in your calculations.) 6. What follows is a way of finding the vector s. If A is a square matrix and v is a nonzero vector such that Av-, then u is called an eigenvector for A with eigenvalue l. If we wish to find u, we can solve the equation Au = u by rewriting it as Au-u = 0, and then as (A-1)t-0. Now we simply need to solve the homogeneous system (A- I)v0. (You know how to do this! Don't be thrown off by the fact that our matrix is now A - I instead of Just A.) Use this idea to find a vector u such that Au = u, where A is the matrix A=10