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Dawson College
Mathematics Department
Final Examination
Linear Algebra
201-105-DW
Sections 01, 02, 03, 04, 05
May 23, 2012
9:30 am-12:30 pm
Student Name: _______________________________________
Student ID: _______________________________________
Teachers: G. Honnouvo, A. Jimenez, V. Ohanyan
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 space provided
Translation and regular dictionaries are permitted Small, noiseless and non-programmable calculators are permitted
This examination consists of 15 pages.
Please ensure that you have a complete examination before starting.
This exam must be returned intact.
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1a) (5 marks) Solve the following linear system.
1 2 3 4
1 2 3 4
1 2 3 4
x x x x
x x x x
x x x x
1b) (2 marks) Find a particular solution where x 2 12
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- (3+3+4 marks) Given
A ,
B ,
C.
a) Find tr ACB 2 I 2
b) Finddet CB
c) Find the matrix X such that X I A
T
1 (^22)
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- (4+3 marks) Given
A ,
b ,
z
y
x
X.
a) Find adj A
b) Use adj A to solve the system AX b.
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7. Let A and B be 2 2 matrices, where det A 2 , det B 3. Find
a) (2 marks)
(^12) det 2 A B
b) (2 marks)det 2 A 7 adj A
1
c) (3 marks)
3 2 det B A 2 BA
T
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- (4+4 marks) Let u 1 , 2 , 3
, v 1 , 2 , k
, w 3 , 1 , 2
.
a) For which values of k the vectors u v
and u v
are perpendicular
b) Find
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11a) (3 marks)) Find the line of intersection of the planes x y z 1 and 2 x y 3 z 1.
12b) (3 marks) Find the parametric equations of the line passing through the point A 2 , 1 , 1
and parallel to the line of intersection of the planes x y z 1 and 2 x y 3 z 1 in part a)
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- (4 marks) Find the equation of the plane containing the lines
z t
y t
x t
and
z s
y s
x s
1
- (4 marks) Find the intersection of the plane 2 x y 2 z 11 and the line
z t
y t
x t
where t is a real number.
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- (4 marks) Find the distance between the point A 2 , 1 , 1 and the line
z t
y t
x t
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- (8 marks) Maximize the profit function P 15 x 1 50 x 2 45 x 3 subject to
1 2 3
1 2 3
1 2 3
1 2 3
where x x x
x x x
x x x
x x x
