CC0678A Semester 2 2015
The University of Sydney
School of Mathematics and Statistics
MATH1014
Introduction to Linear Algebra
November 2015 Lecturers: J Parkinson, B Roberts
Time Allowed: One and a half hours
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 25 questions; the questions are of equal value;
all questions may be attempted.
Answers to the Multiple Choice questions must be entered on
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.
Approved non-programmable, non-graphics calculators may be used.
THE QUESTION PAPER MUST NOT BE REMOVED FROM THE
EXAMINATION ROOM.
Marker’s use
only
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There are three questions in this section.
1. (a) Let A = (6,−1, 1), B = (3, 2, 1) and C = (4, 0,−1).
(i) Find expressions for the vectors
−→
AB,
−→
AC and
−−→
BC.
(ii) Show that ∆ABC is a right angled triangle.
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(b) The line L1 passes through the point (7,−3,−4) and has parametric scalar equations
x = 7 + 3s
L1 : y = −3 + s (s ∈ R).
z = −4 − s
The line L2 passes through the point (7,−3,−4) and has parametric scalar equations
x = 7 − t
L2 : x = −3 + t (t ∈ R).
z = −4 + 2t
(i) Write down vectors v1 and v2 in the direction of L1 and L2 respectively.
(ii) Find the Cartesian equation of the plane that contains the lines L1 and L2.
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(c) (i) Evaluate 4365 in Z7.
(ii) Solve the equation 3x+ 4 = 1 in Z9.
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2. (a) (i) Write down the augmented matrix for the following system of equations, and
reduce the matrix to row echelon form.
x + y − 2z = 5
x + 2y + z = 4
−x + y + 8z = −7
(ii) Write down the general solution of the above system of equations.
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(b) Let A be the matrix

1 −1 02 −1 1
1 −1 1

.
(i) Find the inverse of A.
(ii) Hence (or otherwise) solve the following system of linear equations.
x − y = 1
2x − y + z = 1
x − y + z = −2
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(c) Let B =
[
−5 4
−4 3
]
and C =
[
2 1
3 2
]
.
(i) Calculate BC and CB.
(ii) Find B−1.
(iii) Suppose X is a 2 × 2 matrix such that XB = BC. Does this imply that
X = C?
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3. (a) The females of a colony of beetles live for a maximum of 2 years. In her first year,
each female produces an average of 0.5 female offspring. In her second year, each
female produces an average of 1 female offspring. Half of the females survive their
first year to breed in the second year.
(i) Write down the Leslie matrix for the female beetle population.
(ii) Calculate the eigenvalues of L. For each eigenvalue, find a corresponding
eigenvector.
population for this colony of beetles.
(iv) What are the relative proportions of females in their first and second years in
this stable population?
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(b) Each day for exercise I either ride my bike, swim, or run. I prefer to leave the decision
of my particular form of exercise up to chance, and so I organise my exercise schedule
as follows:
If I ride my bike on a given day then:
with probability 0.9 I will ride my bike again the next day,
with probability 0.1 I will swim the next day, and
with probability 0.0 I will run the next day.
If I swim on a given day then:
with probability 0.5 I will ride my bike the next day,
with probability 0.3 I will swim again the next day,
with probability 0.2 I will run the next day.
If I run on a given day then:
with probability 0.1 I will ride my bike the next day,
with probability 0.8 I will swim the next day,
with probability 0.1 I will run again the next day.
(i) Write down the transition matrix for the Markov chain that models my exer-
cise habits.
(ii) If I run on Monday, what is the probability that I will swim on Wednesday of
the same week?
(iii) Find the steady state probability vector for this Markov chain.
(iv) In the long run, approximately how many days per year will I ride my bike?
(Assume that a year has 365 days).
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This is the last page of the question paper.