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Math Help - Application involving matrices!!!

  1. #1
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    Question Application involving matrices!!!

    So I was going through the old exam papers and found this question. I tried to look for a something a bit similar to this but found nothing. It was hinted that a similar question might appear in this year's exam so if anybody can help me then it'd be greatly appreciated. The question is as follows:

    A study of a colony of a rare sea bird, the fairy tern, indicates that the population changes on a yearly basis as a discrete dynamic system. Dividing the population into 2 classes, juveniles and adults, there are initially 40 juvenile chicks and 20 breeding adults. The system is defined by: (1) an initial state vector x0 = (40 j
    20) a

    and (2) a transition matrix
    A = (0 1.1
    0.5 0.6)

    and (3) the relationship x(n+1) = A*x(n)
    (a) Calculate the state vector x1.
    (b) Which entry in the transition matrix gives the annual birthrate of chicks per adult?
    (c) Which entry of A tells us the probability a chick will survive to become a breeding adult?
    (d) Find the eigenvalues n' corresponding eigenvectors for matrix A.
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  2. #2
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    Quote Originally Posted by Skelly View Post
    So I was going through the old exam papers and found this question. I tried to look for a something a bit similar to this but found nothing. It was hinted that a similar question might appear in this year's exam so if anybody can help me then it'd be greatly appreciated. The question is as follows:

    A study of a colony of a rare sea bird, the fairy tern, indicates that the population changes on a yearly basis as a discrete dynamic system. Dividing the population into 2 classes, juveniles and adults, there are initially 40 juvenile chicks and 20 breeding adults. The system is defined by: (1) an initial state vector x0 = (40 j
    20) a

    and (2) a transition matrix
    A = (0 1.1
    0.5 0.6)

    and (3) the relationship x(n+1) = A*x(n)
    (a) Calculate the state vector x1.
    Multiply x1 = A x0.

    (b) Which entry in the transition matrix gives the annual birthrate of chicks per adult?
    (c) Which entry of A tells us the probability a chick will survive to become a breeding adult?
    Using x1 = A x0, interpret x1 = [chicks, adults] output and x0 = [chicks,adults] input.

    (d) Find the eigenvalues n' corresponding eigenvectors for matrix A.
    The eigenvalues are the roots of characteristic polynomial and the eigenvectors are the corresponding solutions to the characteristic equation.
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