AMS 201.2FIRST Testprof. Tucker Fall, 1992
AMS 201.2FIRST TESTProf. Tucker Fall, 1992
1. Let 1 = [1, 1, 1], e1= [1, 0, 0], e2= [0, 1, 0], e3= [0, 0, 1], I =
Evaluate: a) Ae1, b) e3AI, c) A1, d) (e1A)(Ae2).
2. A rabbit has three different moods each day: Perky, Mopey, and Irritable. If Perky today, it is equally likely to Perky, Mopey or Irritable tomorrow. If Mopey today, it is always Perky tomorrow. If Irritable today, then it has a 1/2 probability of being Perky tomorrow, a 1/6 probability of being Mopey tomorrow, and a 1/3 probability of being Irritable tomorrow.
a) Give the transition matrix for this Markov chains.
b) If Perky today, what are the chances that it is Perky in two days.
3. a) Write the matrix C in partitioned form (give names for submatrices). C =
b) Write C2 in partitioned form. Then write out all the entries in C2.
c) Draw the graph whose adjacency matrix is C.
4. a) What is the sum and max norm of the matrix A in Example 1.
b) If p' = Ap and p" = A2p, and if |p|s= 4, give a bound on the sum norms of p' and of p".
5. Suppose we are given the following matrices: matrix A gives the amounts of raw material required to build different products; matrix B gives the costs of these raw materials in two different countries; and matrix C tells how many of the products are needed to build two types of houses.
Raw Material Cost by Country
A Wood Labor Steel B Spain Italy Items needed in house
Item A 5 10 10 Wood $2 $3 C A B
Item B 4 12 8 Labor $6 $5 House I 4 8
Steel $3 $4 House II 5 5
a) Compute the first row in the matrix product CA (JUST THE FIRST ROW).
b) Which matrix expression tells the cost of each item in each country (the cost of an item is the cost of its raw materials. (DO NOT MULTIPLY OUT)
c) Which matrix expression gives the total cost of building each type of house in each country. (DO NOT MULTIPLY OUT)
6. The matrix A = has eigenvectors u = [1, 2] and v = [1, -1].
a) Determine the two eigenvalues, one for u and one for v.
b) Determine A3v, where v = [5,4], using the eigenvector approach in the text.
c) Give an approximate value for Anv.
AMS 201.2PRACTICE B for FIRST TESTProf. Tucker
1. Let 1 = [1, 1, 1], e1= [1, 0, 0], e2= [0, 1, 0], e3= [0, 0, 1], I , B
Evaluate: a) e1B b) I(Be2), c) B1, d) (e3B)(Be1).
2. If the stock market went up today, tomorrow it is up 50% of the time and either unchanged or down 25% each. If unchanged today, tomorrow it is either up or down 50% each. If down today, tomorrow it is 50% of the time and either up or unchanged 25% each.
a) Give the transition matrix A for this Markov chain.
b) If unchanged today, what are the chances that it is up in two days?
3. Suppose we are given the following matrices: matrix A gives the amounts of seconds each type of computer job takes for the three computer activties; matrix B gives the costs of these activities under different pricing plans; vector c tells how many jobs of each type; and vector d tells the fraction of time each charging plan is used each day.
Time Time Charges Number
A I/O Execute System B Plan I II c of Jobs d Fraction
Job A 5 20 10 I/O 2 3 Job A 4 of Time B
Job B 4 25 8 Exec. 6 5 Job B6 Plan I .3
Job C Syst 3 4 Job C 3 Plan II .7
a) Compute the Bd. What information does it give.
b) Which product gives the total cost of each type of job for each charge plan? (DO NOT EVALUATE)
c) Which product gives the total cost of all jobs when run under plan I and plan II (DO NOT EVALUATE)
4. a) Write the matrix C in partitioned form (give names for submatrices). C
b) Write C2 in partitioned form. Then write out all the entries in the first three rows of C2.
c) Draw the graph whose adjacency matrix is C.
5. a) What is the sum and max norm of the matrix B in Example 1.
b) If x' = Bx and x" = B2x, and if |x|s= 6, give a bound on the sum norms of x' and of x".
6. The matrix A =has eigenvectors u = [2, 1] and v = [1, 1].
a) Determine the two eigenvalues, one for u and one for v.
b) Determine A2v where v = [5,3], using the eigenvector approach in the text.
c) Give an approximate value for Anv.
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