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Dawson College
Mathematics Department
FINAL EXAMINATION
201-NYC-05 section 08
December 17, 2009 9:30- 12:
Teachers: Jacqueline Klasa (substitute teacher Keira Ameur)
Student Name: ___________________________________
Student I.D. #: ____________________________________
Time: 3 Hours
Instructions:
- Print your name and student ID number in the space provided – above.
- All questions are to be answered directly on the examination paper in the provided space. Show your complete work and give explanations.
- Translation and regular dictionaries are permitted.
- Non-programmable calculators are permitted.
Please ensure that you have a complete examination before starting. This exam must be returned intact.
Problem 1.- (15 marks) Solve the following systems of linear equations using the method of your choice: either the Gauss-Jordan method or the Cramer’s rule (only if applicable). For each consistent system, don’t forget to tell how many solutions you found? Whenever you find infinitely many solutions give also a particular solution.
(i) 3x – 7y = 0 6x – 14y = 12
(ii) x - 2y + 3z = 7 2x - 3y = 5 x - 3y + 2z = -
Problem 2.- (12 marks)
Evaluate the determinants of the following matrices and conclude about their invertility:
A =
10 4
15 6
B =
1 1 1
2 0 1
1 5 0
C =
0 0 5
0 0 1
1 2 3
D =
−
− 5 2 0 1
2 1 5 4
4 0 0 0
1 2 0 5
Problem 3.- (12 marks).- Invert the following matrix A using both well known methods:
A =
− 4 4 2
1 0 2
1 3 5
a) Using elementary row-operations.
b) Using cofactors and the adjoint of A.
Problem 4.- (10 marks)
Given the following matrices A, B and C
A =
4 5 6
1 2 3 B =
0
1
0
7
C =
7 1
5 1
3 1
Calculate the following expressions (if possible): 5A , A^2 , AT, AB, AC, A 2 C 2 , (AC)^2 , A + C, BTB, adj(A).
Problem 5.- (15 marks) Here A, B and C represent square matrices of same size.
a) Simplify the following expressions:
(i) (5 B-1^ B^10 C -1^ A ) –1^ (ii) (5 B -1^ (B T^ ) 2 AT^ )T
b) Expand the following expression: (A + B)( I + A -1)(I + B-1) where I is the identity matrix of same size as A and B.
c) If A and B are 4 x 4- matrices with det(A) = -5 and det(B) = 10 , evaluate the following determinants:
(i) det(BA) =
(ii) det(A-1^ B^2 A),
Problem 6.- (12 marks)
Let A, B and C be three points in the space R^3 described by: A = (5,1,-2) B = (2,1,2) and C = (9,1,1).
a) Show that the triangle ABC is a right triangle. Specify where is the right angle. b) Find the coordinates of the midpoint of the side [BC]. c) Write the equation of the plane P containing the triangle ABC. d) Let D be the point equal to (0,5,0). Find the volume of the parallepiped spanned by the 3 vectors: A B
r , A C
r and A D
r .
Problem 8.- (12 marks)
Here is the diagram of the traffic network around the Courthouse square of a city:
A (^) f1 B
f
D f3 C
f
5
10 5
70
3 0
1 0
1 00
a) Write the system of linear equations that represents this traffic network. b) Find all possible flows. c) Because of a parade on the section on South Street along the Courthouse Square, the traffic is closed there (f4 = 0). Describe all resulting flows. d) In normal times (no parades) what are the minimum and maximum flows.
