Semester 1, 2017
The University of Sydney
School of Mathematics and Statistics
MATH1002
Linear Algebra
June 2017 Lecturers: B. Armstrong, D. Badziahin, A. Casella, T.-Y. Chang, J. Ching,
R. Haraway, V. Nandakumar, A. Thomas
Time Allowed: Writing - one and a half hours; Reading - 10 minutes
Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SID: . . . . . . . . . . . . . Seat Number: . . . . . . . . . . . . .
This examination has two sections: Multiple Choice and Extended Answer.
The Multiple Choice Section is worth 50% of the total examination;
there are 20 questions; the questions are of equal value;
all questions may be attempted.
Answers to the Multiple Choice questions must be entered on
the Multiple Choice Answer Sheet.
The Extended Answer Section is worth 50% of the total examination;
there are 3 questions; the questions are of equal value;
all questions may be attempted;
working must be shown.
THE QUESTION PAPER MUST NOT BE REMOVED FROM THE
EXAMINATION ROOM.
Marker’s use
only
Page 1 of 24
Semester 1, 2017 Page 2 of 24
Multiple Choice Section
In each question, choose at most one option.
Your answers must be entered on the Multiple Choice Answer Sheet.
1. The cosine of the angle between the vectors i+ 2j+ 2k and 3i+ 4k is equal to
(a)
14
225
(b)
11
15
(c)
11√
15
(d)
15
11
(e)
225
11
2. If u = −i+ 2j+ 2k and v = 3i− 2j+ k then u · v is equal to
(a) 6i+ 7j− 4k (b) −5 (c) 6i− 7j+ 4k
(d) 9 (e) −9
3. The unit vector in the direction of 2i− j− 2k is
(a)
1
6
(2i− j− 2k) (b) 1
9
(2i− j− 2k) (c) 1
3
(2i− j− 2k)
(d) i+ j+ k (e) i− j− k
4. If a = −i+ j+ k and b = i− j+ k then a× b is equal to
(a) the vector 0 (b) −2i− 2j (c) the scalar 0
(d) 2i+ 2j (e) i− j− 2k
5. Let v = 2i + 6j− k and w = −3i + αj + 3
2
k. Find the value of α so that v and w are
parallel.
(a) 9 (b) −9 (c) −2
3
(d) −3
2
(e) 0
Semester 1, 2017 Page 3 of 24
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
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6. Which one of the following is true for all linearly independent vectors v and w?
(a) v and w are orthogonal.
(b) v and w are parallel.
(c) If av + bw = cv + dw, where a, b, c, d ∈ R, then a = c and b = d.
(d) The vectors v +w and v −w are linearly dependent.
(e) None of the above.
7. The system of equations
2x − 2y − 2z = 4
x − 5z = 0
5x − 4y − 9z = 8
has the general solution
(a) x = 5, y = 2, z = 1.
(b) x = 0, y = −1, z = 0.
(c) x = 5t, y = 4t− 2, z = t where t ∈ R.
(d) x = 1 + 2t, y = 2 + 9t, z = t where t ∈ R.
(e) x = 5t, y = 4t+ 2, z = t where t ∈ R.
8. Let A = P D1 P
−1, and B = P D2 P−1 where D1 =
[
1 0
0 2
]
, D2 =
[
3 0
0 1
]
, and
P =
[
1 1
1 0
]
. Then (AB)5 is
(a)
[
25 35 − 25
0 35
]
. (b)
[ −35 25 − 35
0 −25
]
. (c)
[
35 0
0 25
]
.
(d)
[
25 1
0 35
]
. (e)
[ −25 −35 + 25
0 −35
]
.
9. The determinant of the matrix
 2 2 03 −1 2
1 2 −1
 is equal to
(a) −4 (b) 4 (c) 16 (d) −16 (e) 0
10. Let B be a 4× 4 matrix and suppose that det(B) = 2 . Then det
(
− 1√
2
B
)
is
(a) 1/2 (b) −1/
√
2 (c) −1 (d) −
√
2 (e) −2
Semester 1, 2017 Page 5 of 24
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
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11. Which one of the following augmented matrices is in row echelon form?
(a)
 1 0 20 0 0
0 0 1
 (b)
 1 −3 0 −11 6 1 4
0 0 0 2
 (c) [ 0 0 1 2
1 1 0 1
]
(d)
[
1 3 2
0 0 1
]
(e) None of the above.
12. Which one of the following is true for all n× n matrices A and B?
(a) (A+B)2 = A2 + 2AB +B2
(b) if A2 = B2 then A = ±B
(c) if AB = 0 then A = 0 or B = 0
(d) if A is invertible then AB is invertible
(e) if A is invertible and AB is invertible then B is invertible
13. If A =
 1 1 01 1 1
0 1 1
, which one of the following is true?
(a) A is not invertible.
(b) A is invertible and A−1 =
 0 1 11 1 1
1 1 0
.
(c) A is invertible and A−1 =
 1 −1 0−1 1 −1
0 −1 1
.
(d) A is invertible and A−1 =
 0 1 −11 −1 1
−1 1 0
.
(e) None of the above.
14. Which one of the following statements is true, given that A is a matrix of size 3× 3,
B is a matrix of size 3× 2, and C is a matrix of size 2× 3?
(a) A2 +BC is a 3× 3 matrix. (b) ACB is defined.
(c) 2A+ CB is defined. (d) (BC)2 is a 2× 2 matrix.
(e) B(A−BC) is defined.
Semester 1, 2017 Page 7 of 24
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
Semester 1, 2017 Page 8 of 24
15. Consider the three planes with equations
P1 : x + 2y − z = 1,
P2 : −2x − 4y + 2z = 2,
P3 : −3x − 6y + 3z = 6.
Which of the following is true?
(a) None of the planes are parallel to each other.
(b) P1 and P2 are parallel to each other but not parallel to P3.
(c) P1 and P3 are parallel to each other but not parallel to P2.
(d) P2 and P3 are parallel to each other but not parallel to P1.
(e) All of the planes are parallel to each other.
16. Consider the following system of equations:
x + y + z − w = 0
−y + 2z − w = 0
2x + 6z − 4w = 0
Which one of the following statements about this system is true?
(a) There is a unique solution.
(b) The general solution is expressed using exactly 1 parameter.
(c) The general solution is expressed using exactly 2 parameters.
(d) The general solution is expressed using 3 or more parameters.
(e) There is no solution.
17. Which one of the following sequences of row operations, when applied to the matrix[
a b c
d e f
]
, produces the matrix
[
d− a e− b f − c
3a 3b 3c
]
?
(a) First R1 := R1 −R2, then R2 := 3R2, then R1 ↔ R2.
(b) First R1 ↔ R2, then R1 := 3R1, then R1 := R1 −R2.
(c) First R2 := R2 −R1, then R1 ↔ R2, then R1 := 3R1.
(d) First R1 := 3R1, then R1 ↔ R2, then R1 := R1 −R2.
(e) First R1 ↔ R2, then R1 := R1 −R2, then R2 := 3R2.
Semester 1, 2017 Page 9 of 24
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
Semester 1, 2017 Page 10 of 24
18. Suppose v and w are two non-zero vectors lying in this page:
v w
Which of the following is true?
(a) (v ×w) · v is a non-zero scalar.
(b) v and v ×w are parallel.
(c) (v ×w)× v is perpendicular to both v and w.
(d) (w × v)× (v ×w) is parallel to v but not w.
(e) v ×w points upwards, towards the ceiling.
19. The two lines given by the respective parametric scalar equations
x = 3 + t
y = −5 + 2t
z = −5 − t
 t ∈ R and x = −3 − 2sy = −2 − 4sz = 1 + 2s
 s ∈ R
(a) do not intersect. (b) intersect at the point (7, 3,−9).
(c) intersect at the point (−2,−15, 0). (d) intersect at the point (−3,−2, 1).
(e) coincide.
20. Suppose a 3 × 3 matrix A has 3 distinct eigenvalues λ1, λ2 and λ3. Which one of the
following is NOT necessarily true?
(a) The characteristic polynomial of A has 3 distinct roots.
(b) det(A) = λ1λ2λ3.
(c) A is invertible.
(d) There is a 3× 3 invertible matrix P so that PAP−1 =
 λ2 0 00 λ3 0
0 0 λ1
.
(e) If B is any 3× 3 invertible matrix then BAB−1 has eigenvalues λ1, λ2 and λ3.
Semester 1, 2017 Page 11 of 24
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
Semester 1, 2017 Page 12 of 24
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
Semester 1, 2017 Page 13 of 24
This blank page may be used for rough working; it will not be marked.
End of Multiple Choice Section
Make sure that your answers are entered on the Multiple Choice Answer Sheet
The Extended Answer Section begins on the next page
Semester 1, 2017 Page 14 of 24
Extended Answer Section
There are three questions in this section, each with a number of parts. Write your answers
in the space provided. If you need more space there are extra pages at the end of the
examination paper.
1. (a) Let v = 2i− j+ 3k and w = 4i− 3j+ k.
(i) Find the area of the parallelogram inscribed by the vectors v and w.
(ii) Is the parallelogram inscribed by the vectors v and w a rectangle? Either
prove that it is, or prove that it isn’t.
Semester 1, 2017 Page 15 of 24
(b) Let P be the plane with Cartesian equation 3x−4y+z = 5. What are the Cartesian
equations of the line L that is perpendicular to the plane P and passes through the
point Q = (2, 1,−7)?
Semester 1, 2017 Page 16 of 24
(c) Let A =
[
a b
c d
]
and suppose that for all vectors v =
[
v1
v2
]
and w =
[
w1
w2
]
,
(Av) · (Aw) = v ·w.
Prove that if B =
[
a c
b d
]
then BA = I.
[Hint: consider particular vectors v and w.]
Semester 1, 2017 Page 17 of 24
2. (a) A quadratic polynomial P (x) = ax2 + bx + c satisfies P (1) = 1, P (2) = 2 and
P (4) = 3. Find a, b and c.
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(b) Find the value(s) of the parameter a such that the following system of linear equa-
tions is inconsistent.
x+ 2y + z = 1
2x+ 4y + az = 2
x+ 2ay + 2z = −1
Semester 1, 2017 Page 19 of 24
(c) You are given that the matrix M =
 0 −2 18 −15 6
21 −36 14
 satisfies the equation
M3 = −M2 −M − I.
Compute M25.
Semester 1, 2017 Page 20 of 24
3. (a) Let
A =
[
0 4
1 0
]
.
Find a diagonal matrix D so that A = PDP−1 for some invertible matrix P . You
do not need to find the matrices P or P−1.
(b) You are given that the matrix
B =
−1 0 32 −1 5
1 1 0

has λ = −2 as one of its eigenvalues. Find the (−2)-eigenspace of B.
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(c) Let C and D be n× n matrices, with D invertible. Prove that if λ is an eigenvalue
of C, then λ2 is an eigenvalue of D−2C2D2.
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There are no more questions.
More space is available on the next page.
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This blank page may be used if you need more space for your answers.
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This blank page may be used if you need more space for your answers.
End of Extended Answer Section
This is the last page of the question paper.
B Semester 1, 2017 Multiple Choice Answer Sheet
0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

Write your
SID here −→
Code your
SID into
the columns
below each
digit, by
filling in the
appropriate
oval.
Answers −→
a b c d e a b c d e
Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q8

Q9

Q10

Q11

Q12

Q13

Q14

Q15

Q16

Q17

Q18

Q19

Q20

The University of Sydney
School of Mathematics and
Statistics
MATH1002 Linear Algebra
Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Seat Number: . . . . . . . . . . . . . . . . .
Indicate your answer to each question by
filling in the appropriate oval.
This is the first and last page of this answer sheet
Correct Responses to MC Component of
MATH1002 Linear Algebra
: Semester 1, 2017
Q1 −→ b
Q2 −→ b
Q3 −→ c
Q4 −→ d
Q5 −→ b
Q6 −→ c
Q7 −→ c
Q8 −→ a
Q9 −→ b
Q10 −→ a
Q11 −→ d
Q12 −→ e
Q13 −→ d
Q14 −→ a
Q15 −→ e
Q16 −→ c
Q17 −→ e
Q18 −→ e
Q19 −→ a
Q20 −→ c