Semester 1 Main, 2019

The University of Sydney

School of Mathematics and Statistics

MATH1002
Linear Algebra

June 2019 Lecturers: Z. Afsar, A. Aksamit, D. Badziahin, A. Fish, A. Kerschl, B. Pauwels.

Time Allowed: Writing - one and a half hours; Reading - 10 minutes

Exam Conditions: This is a closed-book examination — no material permitted. Writing

is not permitted at all during reading time.

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This examination has two sections: Multiple Choice and Extended Answer. Marker’s use
only

The Multiple Choice Section is worth 50% of the total examination. 1

There are 20 questions. The questions are of equal value.
All questions should be attempted. 2

Answers to the Multiple Choice questions must be entered on

3
the Multiple Choice Answer Sheet before the end of the examination.

The Extended Answer Section is worth 50% of the total examination.

There are 3 questions. The questions are of equal value.
All questions should be attempted. Working must be shown.

THE QUESTION PAPER MUST NOT BE REMOVED FROM THE

EXAMINATION ROOM.

Page 1 of 21
Semester 1 Main, 2019 Page 2 of 21

Multiple Choice Section

For each question, choose exactly one option.
Your answers must be entered on the Multiple Choice Answer Sheet.

1. Let u = [1, α, 0], v = [5, −4, 15] and w = [−1, β, −3]. Find the values of α and β so
that u and v are orthogonal and v and w are parallel.

5 4 4 5 5 4
(a) α = , β = (b) α = , β = (c) α = , β = −
4 5 5 4 4 5
5 4 4 5
(d) α = − , β = (e) α = − , β =
4 5 5 4

2. What is the area of the triangle inscribed by the vectors [2, −1, 3] and [−1, 1, 0]?
√ √
√ 19 7 19 √
(a) 19 (b) (c) (d) (e) 7
2 2 2

3. Which
 one of the following
 augmented matrices is in row echelon form?
1 −3 0 −1    
2 4 3 0 0 1 2
(a) 1 6 1 6
  (b) (c)
0 0 1 1 1 0 1
0 0 0 2
 
1 0 4
(d)  0 0 0  (e) None of the above.
0 1 0

 
1 1 0
4. The determinant of the matrix  3 −1 2  is equal to
2 4 −2
(a) 16 (b) −16 (c) 0 (d) 4 (e) −4
Semester 1 Main, 2019 Page 3 of 21

5. Consider the following system of equations:

x + y + z − w = 0
−y + 2z − w = 0
3z − 2w = 0
Which one of the following statements about this system is true?
(a) There is no 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 a unique solution.

6. 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 perpendicular to both v and w.

(b) (w × v) × (v × w) is parallel to w but not v.
(c) (v × w) · w is a non-zero scalar.
(d) v × w points upwards, towards the ceiling.
(e) w and v × w are parallel.

7. The two lines given by the respective parametric equations

 
x = 3 + t  x = −3 − 2s 
y = −5 + 2t t∈R and y = −2 − 4s s∈R
z = −5 − t  z = 1 + 2s 

(a) do not intersect. (b) intersect at the point (−3, −2, 1).
(c) intersect at the point (7, 3, −9). (d) intersect at the point (−2, −15, 0).
(e) coincide.
Semester 1 Main, 2019 Page 4 of 21

8. The cosine of the angle between the vectors [1, 3, −5] and [3, 2, 1] is equal to
√
7 10 11 2 4 2
(a) (b) (c) √ (d) √ (e)
4 15 7 10 7 10 245

9. Consider the three planes with equations

P1 : x + 2y − z = 1,
P2 : 2x − 5y + 3z = 1,
P3 : 2x − 4y + z = 10.
Which of the following is true?
(a) P1 and P2 are parallel to each other but not parallel to P3 .
(b) P1 and P3 are parallel to each other but not parallel to P2 .
(c) P2 and P3 are parallel to each other but not parallel to P1 .
(d) None of the planes are parallel to each other.
(e) All of the planes are parallel to each other.

10. Which
 oneof the following sequences
 of row operations, when applied to the matrix
a b c d−a e−b f −c
, produces the matrix ?
d e f 2a 2b 2c
(a) First R1 → R1 − R2 , then R2 → 2R2 , then R1 ↔ R2 .
(b) First R1 → 2R1 , then R1 ↔ R2 , then R1 → R1 − R2 .
(c) First R2 → R2 − R1 , then R1 ↔ R2 , then R1 → 2R1 .
(d) First R1 ↔ R2 , then R1 → 2R1 , then R1 → R1 − R2 .
(e) First R1 ↔ R2 , then R1 → R1 − R2 , then R2 → 2R2 .

11. Suppose that v and w are two linearly independent vectors in Rn . Which one of the
following statements is true?
(a) v and w are orthogonal.
(b) If av + bw = cv + dw, where a, b, c, d ∈ R, then a = c and b = d.
(c) v and w are parallel.
(d) The vectors v + w and v − w are linearly dependent.
(e) None of the above.
Semester 1 Main, 2019 Page 5 of 21

12. Which one of the following statements is true?

(a) Any set of three vectors from R2 must span R2 .
(b) There are three vectors u, v, w ∈ R2 such that {u, v, w} is linearly independent.
(c) If span{u, v} = R2 , then {u, v} is a basis for R2 .
(d) For any three vectors u, v, w ∈ R2 , there is a subset of {u, v, w} that is a basis for
R2 .
(e) The set {u, v, 0} is a basis for R2 only if {u, v} is a basis for R2 .

13. 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) ACB is defined. (b) 2A + CB is defined.

(c) (BC)2 is a 2 × 2 matrix. (d) B(A − BC) is defined.
(e) A2 + BC is a 3 × 3 matrix.

 
1 0 0
14. If A =  0 1 3 , which one of the following is true?
0 0 1
(a) A is not invertible.
 
0 1 −1
(b) A is invertible and A−1 =  1 −1 1 .
−3 1 0
 
1 0 0
(c) A is invertible and A−1 = 0 1 −3 .

0 0 1
 
0 1 1
(d) A is invertible and A−1 =  1 1 1 .
−3 1 0
(e) None of the above.
Semester 1 Main, 2019 Page 6 of 21

   
−1 −1 1 0 3 0
15. Let A = P D1 P , and B = P D2 P where D1 = , D2 = , and
0 2 0 1
 
1 1
P = . Then (AB)5 is
1 0
 5 5
2 3 − 25 −35 25 − 35
    5 
3 0
(a) . (b) . (c) .
0 35 0 −25 0 25
 5
−25 −35 + 25
  
2 1
(d) . (e) .
0 35 0 −35

16. Let A and B be n × n diagonalisable matrices. Which one of the following is always
true?
(a) A + B is diagonalisable.
(b) AB is diagonalisable.
(c) A2 and B 2 are diagonalisable.
(d) AB = BA.
(e) none of the above.

 
1
17. Let B be a 4 × 4 matrix and suppose that det(B) = 3 . Then det − √ B is
3
√ √
(a) 1/3 (b) 3 (c) −1/ 3 (d) − 3 (e) −1

18. Suppose a 3 × 3 matrix A has 3 distinct eigenvalues λ1 , λ2 and λ3 . Which one of the
following is NOT necessarily true?

(a) det(A) = λ1 λ2 λ3 .
(b) The characteristic polynomial of A has 3 distinct roots.
(c) A is invertible.
(d) If B is any 3 × 3 invertible matrix then BAB −1 has eigenvalues λ1 , λ2 and λ3 .
 
λ2 0 0
(e) There is a 3 × 3 invertible matrix P so that P AP −1 =  0 λ3 0 .
0 0 λ1
Semester 1 Main, 2019 Page 7 of 21

19. Let P be a transition matrix of a Markov chain on n states. Which of the following is
NOT necessarily true.
(a) P 2 is a transition matrix for a Markov chain.
(b) P is an n × n matrix.
(c) If P is invertible, then P −1 is a transition matrix for a Markov chain
(d) If Q is another transition matrix for a Markov chain on n states, then P Q is a
transition matrix for a Markov chain.
1
(e) If Q is another transition matrix for a Markov chain on n states, then (P + Q) is
2
a transition matrix for a Markov chain

20. Which one of the following is true for all n × n matrices A and B?
(a) if A2 = B 2 then A = ±B
(b) if AB = 0 then A = 0 or B = 0
(c) (A + B)2 = A2 + 2AB + B 2
(d) if A is invertible then AB is invertible
(e) if AB is invertible then A and B are invertible
Semester 1 Main, 2019 Page 8 of 21

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 Main, 2019 Page 9 of 21

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 P1 , P2 and P3 be three planes in three dimensional space given by the equations
P1 : x + 3y + 4z = 5; P2 : 2x + 5y + z = b; P3 : −x + ay + 3z = 0,
where a, b ∈ R are parameters.

(i ) Find n1 × n2 , where n1 is a normal vector for P1 , and n2 is a normal vector

for P2 .

(ii ) Find the direction vector of the line of intersection of P1 and P2 .

Semester 1 Main, 2019 Page 10 of 21

(iii ) Find the parameters a and b such that three planes P1 , P2 and P3 have empty
intersection.
Semester 1 Main, 2019 Page 11 of 21

(b) (i ) Find the projection of the vector [3, 2, 1] onto the line spanned by the vector
[1, 2, 3].

(ii ) By use of part (i) or otherwise, find the distance between the point P (3, 2, 1)
and the line with the parametric equations

x=t 
y = 2t t ∈ R.
z = 3t 
Semester 1 Main, 2019 Page 12 of 21

(c) Show that any three non-zero vectors u, v, w in R3 which are orthogonal one to each
other are linearly independent.
Semester 1 Main, 2019 Page 13 of 21

 
0 1 2
2. (a) Find the inverse of the matrix 1 0 0 .
1 1 1
Semester 1 Main, 2019 Page 14 of 21

(b) Use your answer to (a), or another valid method, to write vectors e2 = [0, 1, 0]
and e3 = [0, 0, 1] as linear combinations of vectors v1 = [0, 1, 1], v2 = [1, 0, 1],
v3 = [2, 0, 1].
Semester 1 Main, 2019 Page 15 of 21

(c) For which value of the parameter a, the set Sa = {[x, y] ∈ R2 | x + axy = 0} is a
subspace of R2 ? Provide a proof of your answer.
Semester 1 Main, 2019 Page 16 of 21

3. (a) Let    
0 0 1/2 1/2
P = 1/2 1 0  and x0 =  0  ,
1/2 0 1/2 1/2
where P is the transition matrix for a Markov chain with three states, and x0 is the
initial state vector for the population.
(i) Compute x1 and x2 .

(ii ) What is the probability of moving from state 3 to state 1 in 2 transitions?

Semester 1 Main, 2019 Page 17 of 21

(iii ) Find lim P n x0 . (You can assume that the limit exists)
n→∞
Semester 1 Main, 2019 Page 18 of 21

(b) Let A and B be n × n matrices.

(i ) Let λ ̸= 0. Show that λ is an eigenvalue of AB if and only if it is also an

eigenvalue of BA.
Semester 1 Main, 2019 Page 19 of 21

(ii ) Show that In + AB is invertible if and only if In + BA is invertible, where In

is the identity n × n matrix.
Semester 1 Main, 2019 Page 20 of 21

There are no more questions.

More space is available on the next page.

Semester 1 Main, 2019 Page 21 of 21

This blank page may be used if you need more space for your answers.

End of Extended Answer Section.

End of Examination
 Semester 1 Main, 2019 Multiple Choice Answer Sheet

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