Math 210 Fall 2008  

Problem Set 8, Due: Tuesday, Nov. 13, 2008 (in class)
(late papers OK until 1:00 Friday)

The following sections Eigenvalues and Eigenvectors from Strang's Linear Algebra and its Applications may be helpful.
  1. You are given an invertible matrix A. Say V is an eigenvector with eigenvalue 7, so AV = 7V.
    Compute   A2V + A-1V.

  2. Find the eigenvalues and eigenvectors of the matrix 
                               | 2  5 |
                       A = |      |
                           | 5  2 |

  3. Let A be the transition matrix of a Markov chain and V be an eigenvector with eigenvalue not 1, show that the sum of the components of V is zero.

  4. Find the eigenvalues and eigenvectors of the 3 × 3 matrix all of whose elements are ones.
    Repeat this for the 3 × 3 matrix all of whose elements are 1/3.

    A Computer Problem

  5. Susan borrows P0 dollars to buy a car. The annual interest rate is i (perhaps 6%, 7%, etc.) compounded monthly, so at the end of the first month she owes [1 + (i /12)]P0 dollars, where here i is written as, say, .06, etc.. She will repay the loan with equal monthly payments of M dollars. Thus, just after she has made the first monthly payment she owes
    P1 = [ 1 + (i /12) ]P0 - M
    dollars. One month later, she owes [1 + (i /12)]P1 dollars so just after she has made the second monthly payment she owes
    P2 = [ 1 + (i /12) ]P1 - M
    and so on. One can also write this as
    M = (P1 -P2) + I2
    where (P1 -P2) in the decrease in the amount owed and
    I2 = (i /12) P1
    is the interest component of the second monthly payment.
    a). What is the minimum monthly payment M if her balance is to decrease?
    b). Write the formula for P2, the amount she owes just after making the second monthly payment, in terms of P0 and M. [Remark: I suggest letting c = 1 + (i /12). It makes the computation more transparent.]
    c). How much, Pk, does she owe just after making the kth monthly payment? [Write your answer in terms of P0, M, and k. You may find it useful to recall the formula for the sum of a geometric series: 1+r+r2+...+rn].
    d). Give a formula for the interest component of the kth monthly payment   Ik = (i /12)Pk-1.  Your formula should involve only P0, M, i (or c), and k.
    e). How many monthly payments, N, are needed until the loan is completely repaid? [Hint: take logarithms].
    f). If the loan is to be repaid in exactly N monthly payments, how much, M, should she repay each month?
    g). Write a web script (running on johnny.sas) that does this computation for someone. Thus on a web form, you ask the user to input the amount of the loan (P0), the annual interest rate (i) and either M or N. You tell them either N or M (whichever they wanted to compute).
    Remark: For tax records, one should also give the amount of interest I k contained in the kth monthly payment, but don't do that here since this problem is already long enough.