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Matrix problem

  1. Oct 26, 2010 #1
    1. The problem statement, all variables and given/known data
    Suppose that a country is divided into three regions: Upper, Lower and Central. Each year, one-quarter of the residents of the Upper region move to the Lower region and the remaining residents stay in the Upper region. One-half of the residents of the Lower region move to the Central region and the remaining residents remain in the Lower region. Three-quarters of the residents of the Central region move to the Lower region, and the remaining residents stay in the Central region.
    In the long run, what proportion of the residents settle in each region?
    of the total residents settle in the Upper region,
    of the total residents settle in the Lower region, and
    of the total residents settle in the Central region.


    2. Relevant equations



    3. The attempt at a solution
    I have
    U=3/4U
    L=1/4 U+1/2L+3/4C
    C=0 +1/2L+1/4C
    I don't know what to do next ,please help..
     
  2. jcsd
  3. Oct 26, 2010 #2

    Dick

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    Express that linear system in matrix form. (U,L,C)=M(U,L,C) where M is a 3x3 matrix. What's M? If there is a steady state then M has a eigenvalue of 1. What's the corresponding eigenvector?
     
  4. Oct 26, 2010 #3
    M is
    3/4 0 0
    1/4 1/2 3/4
    0 1/2 1/4
    What do u mean "steady state" and "eigenvalue"? I don't think I learn these yet...
    btw, is there a simple way to do it?
    ty
     
  5. Oct 27, 2010 #4

    Dick

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    Yes, actually, there is. Just solve the equations you have for U, L and C. What does the first equation tell you about U?
     
  6. Oct 27, 2010 #5
    what about the total population, do i assume 1? or 3?
    If 3,
    then i got 25%=U , 50%= L, 25%=C
    but I don't know how to determine the long run, but I think U is going to be 0
     
  7. Oct 27, 2010 #6

    Dick

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    I hope you concluded U=0 from one of your equations. You don't have to assume anything for the total population. Call it P. So U+L+C=P. Or put P=1 if you just want to work with percentages.
     
  8. Oct 27, 2010 #7
    no, I just guessing, cause i tried to cube the matrix and multiply it by 1, so in the third year ,U is decreasing, L and C is increasing
    but how do i get the long run?
     
  9. Oct 27, 2010 #8

    Dick

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    If you don't know 'eigenvalue' and 'eigenvector' forget about the matrix. Just solve the original equations you wrote down.

    U=3/4U
    L=1/4 U+1/2L+3/4C
    C=0 +1/2L+1/4C

    As I said before, what does the first equation tell you about U?
     
  10. Oct 27, 2010 #9
    the final population of Uf is 3/4 of the initial population(Ui)?
     
  11. Oct 27, 2010 #10

    Dick

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    That's true if you mean U_i is the population at the beginning of the year and U_f is the population at the end of the year. The problem says "In the long run". They are implying that the population of each area will settle down to a constant value. So that equation becomes U=(3/4)U.
     
    Last edited: Oct 27, 2010
  12. Oct 27, 2010 #11
    if constant,do you mean the answer for U is 3/4?
     
  13. Oct 27, 2010 #12

    Dick

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    Does U=3/4 satisfy U=(3/4)U?
     
  14. Oct 27, 2010 #13
    no...
    sorry , i still don't get it..
     
  15. Oct 27, 2010 #14

    Dick

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    U=(3/4)U. Solve for U. Use algebra. How would you do that?
     
  16. Oct 27, 2010 #15
    1=3/4?
    that's what i am confusing
    or do you mean Uf/Ui=3/4? Uf=4 Ui=3?
     
  17. Oct 27, 2010 #16
    oh, I think I got it....
    U=0
    0=(3/4)*0?
     
  18. Oct 27, 2010 #17

    Dick

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    That's it. U=0. Now put that into the other equations and find a relation between L and C.
     
  19. Oct 27, 2010 #18
    ok ,i got 60% for L and 40% for C
    But I don't understand why L=L,C=C in the long run...
     
  20. Oct 27, 2010 #19

    Dick

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    Suppose U=0, L=60 and C=40 in one year. Following the instruction in the problem what are U, L and C in the next year?
     
  21. Oct 27, 2010 #20
    the answer is same,
    I see..
    what if U is not equal to zero,
    like U=3/4U+1/2L
     
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