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 consists of 12 pages, numbered from 1 to 12.
There are 3 questions, numbered from 1 to 3.
Marker’s use
only
Page 1 of 12
Semester 1, 2017 Page 2 of 12
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 = 2e1 − e2 + 3e3 and w = 4e1 − 3e2 + e3.
(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 3 of 12
(b) Let P be the plane with general equation 3x−4y+ z = 5. What are the parametric
equations of the line L that is perpendicular to the plane P and passes through the
point Q = (2, 1,−7)?
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(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.]
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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
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(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.
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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.
Semester 1, 2017 Page 11 of 12
This blank page may be used if you need more space for your answers.
Semester 1, 2017 Page 12 of 12
This blank page may be used if you need more space for your answers.
This is the last page of the question paper.

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