Assignment 6, MATH 240, Spring 2008.
 Due 9:30am, Wednesday March 26th.
 Please post your solutions in the drop-off box outside the lab AQ 4135.
 You don't need to hand in solutions to odd numbered questions.
 Sorry, no late assignments accepted again.
 
 From the text Fraleigh & Beauregard
 9.1 exercises 2,6,7,18,20,21
 5.1 exercises 8,16,23,28,36
 5.2 exercises 2,10,13,14,18
 
 Additional exercises:
 
 1: Calculate the eigenvalues and eigenvectors of A, B, and C below.
    See the handout with plots of the eigenvectors of B and C.
    Note, A has complex eigenvalues but B and C have real eigenvalues.
 
        [ 1  -1 ]          [ 1   1 ]           [ 1   1 ]
    A = [       ]      B = [       ]       C = [       ]
        [ 1   1 ]          [ 1   0 ]           [ 0   1 ]
 
 2: Suppose we model a population growth using three age groups.
    Let F1, F2, F3 be the fertility rates and let P1, P2, and P3 be the
    survival rates of the three age groups.
    Suppose F1=0, F2=1, F3=1, P1=1/2, P2=2/3, P3=2/3, that is
    So the Leslie matrix is
 
                [  F1   F2   F3  ]    [   0    1    1  ]
            L = [  P1    0    0  ]  = [  1/2   0    0  ]
                [   0   P2   P3  ]    [   0   2/3  2/3 ]
 
    Calculate the eigenvalues of this matrix L.  Is the population growing
    or declining?  What is the long term population distribution vector?

    P1, the survival rate of the first age group, is 1/2.
    What must P1 be for the population to stabilize?
    What is the long term population distribution vector in that case?

    F2, the fertility rate of the second age group, is 1.
    What must F2 be for the population growth to stabilize?

 3: (optional)
    Let A be an n by n matrix and let M(n) be the number of multiplications of
    real numbers done by formula (3) on page 252 for computing det(A).
    In class we showed  M(1) = 0, M(2) = 1, and M(n) = n + n M(n-1) for n > 1.
    Calculate M(3), M(4), M(5), M(6), M(7), M(8), M(9), M(10) exactly.
    Calculate M(n)/n! in decimal and observe that M(n) > n! for n>2 .
    Can you guess what the limit of M(n)/n! as n goes to infinity is?
 
 Notes on exercises.
 
 For exercise 18 of 9.1, finding the three cube roots of 8, i.e. solving
 x^3 = 8 is equivalent to finding the roots of the polynomial x^3-8.
 One root is x=2 which is real.  The other two are complex.  To find them
 first divide x^3-8 by x-2 using polynomial long division to remove the root x=2.
 
 For exercises 18 of 5.2, hint: if the characteristic polynomials
 of two matrices are the same then their eigenvalues must be the same.