University of Ottawa
Department of Mathematics and Statistics
MAT 1302A: Mathematical Methods II
Professor: Aziz Khanchi
Final Exam – Solutions
April 2010
Surname
First Name
Student #
Seat #
Instructions:
(a) You have 3 hours to complete this exam.
(b) This exam consists of two parts. Questions 1 through 14 are
answer only
. For these
questions, only your final answer will be considered for marks. Questions 15 through
22 are
long answer
. For these questions, you must show your work and justify your
answers to receive full marks.
Partial marks may be awarded for making sufficient
progress towards a solution.
(c) Each answer only question is worth two points.
For the long answer questions, the
number of points available for each question is indicated in square brackets.
(d) All work to be considered for grading should be written in the space provided.
The
reverse side of pages is for scrap work. If you find that you need extra space in order to
answer a particular question, you should continue on the reverse side of the page and
indicate this
clearly
. Otherwise, the work written on the reverse side of pages will not
be considered for marks.
(e) Write your student number at the top of each page in the space provided.
(f) No notes, books, scrap paper, calculators or other electronic devices are allowed.
(g) You may use the last page of the exam as scrap paper.
Good luck!
Please do not write in the table below.
Question
1–14
15
16
17
18
19
20
21
22
Total
Maximum
28
4
4
4
6
8
6
6
4
70
Grade
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MAT 1302A Final Exam – Solutions
Part A: Answer Only Questions
For Questions 1–14, only your final answer will be considered for marks. Each question is worth
2 points.
Question 1.
What is the determinant of the matrix
4
7
10

4
0
0
1
/
2

6
7
8
0
0
3
4

5
/
2
0
0
0

2
7
/
3
0
0
0
0
1
?
Answer:

12
Question 2.
Give the characteristic polynomial of the matrix
2
0
0
0

5
1
0
0
7
0
2
0
5

6
3
0
.
Answer:
λ
(
λ

1)(
λ

2)
2
Question 3.
Suppose
A
=
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
,
and det
A
= 3. If
B
=
a
21
a
22
a
23
a
11
a
12
a
13
a
31
a
32
a
33
,
and
C
=
a
11
a
12
a
13
a
21
a
22
a
23
a
31

3
a
11
a
32

3
a
12
a
33

3
a
13
,
what are det
B
and det
C
?
Answer:
det
B
=

3, det
C
= 3
Question 4.
Is
1 +
i
1
an eigenvector of the matrix

1
2

1
1
? If so, what is the corre
sponding eigenvalue?
Answer:
Since

1
2

1
1
1 +
i
1
=
1

i

i
=

i
1 +
i
1
,
the answer is YES and the eigenvalue is

i
.
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