Student
number
Semester 1 Assessment, 2021
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School of Mathematics and Statistics
MAST10008 Accelerated Mathematics 1
Reading time: 30 minutes — Writing time: 3 hours — Upload time: 30 minutes
This exam consists of 17 pages (including this page)
Permitted Materials
• This exam and/or an offline electronic PDF reader, one or more copies of the masked
exam template made available earlier and blank loose-leaf paper.
• One double sided A4 page of notes (handwritten or printed).
• No calculators are permitted.
Instructions to Students
• If you have a printer, print the exam one-sided. If using an electronic PDF reader to read
the exam, it must be disconnected from the internet. Its screen must be visible in Zoom.
No mathematical or other software on the device may be used. No file other than the
exam paper may be viewed.
• Ask the supervisor if you want to use the device running Zoom.
Writing
• There are 12 questions with marks as shown. The total number of marks available is 87.
• You should attempt all questions.
• Write your answers in the boxes provided on the exam that you have printed or the masked
exam template that has been previously made available. If you need more space, you can
use blank paper. Note this in the answer box, so the marker knows. The extra pages can
be added to the end of the exam to scan.
• If you have been unable to print the exam and do not have the masked template write
your answers on A4 paper. The first page should contain only your student number, the
subject code and the subject name. Write on one side of each sheet only. Start each
question on a new page and include the question number at the top of each page.
Scanning
• Put the pages in number order and the correct way up. Add any extra pages to the end.
Use a scanning app to scan all pages to PDF. Scan directly from above. Crop pages to
A4. Make sure that you upload the correct PDF file and that your PDF file is readable.
Submitting
• You must submit while in the Zoom room. No submissions will be accepted after
you have left the Zoom room.
• Go to the Gradescope window. Choose the Canvas assignment for this exam. Submit
your file. Wait for Gradescope email confirming your submission. Tell your supervisor
when you have received it.
©University of Melbourne 2021
Page 1 of 17 pages
Can be placed in Baillieu Library
MAST10008 Accelerated Mathematics 1
Semester 1, 2021
Question 1 (6 marks)
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Let A, B, C, E be n × n matrices. Suppose that det(A) = 2, det(B) = 4, det(C) = 5.
(a) Calculate:
(i) det(C 10 AT B −1 C −9 )
(ii) rank(2ABC)
(b) Show that if n is odd and E T = −E, then det(E) = 0.
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 2 (8 marks)
Let a ∈ R be a parameter. Consider the plane E with cartesian equation x − 2y + az = 6, and
the line L with vector form (x, y, z) = (1, −1, 0) + t(−3, −a, a).
(a) For which values of a does the origin lie on the line L?
(b) For which values of a do L and E intersect?
(c) Find the point of intersection if L and E do intersect.
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 3 (9 marks)
For each of the following subsets of the indicated vector space, determine if it is a subspace.
If it is a subspace, prove it. If it is not, give a counterexample to one of the conditions of the
subspace theorem.
(a) W1 = {(x, y, z) ∈ R3 |xy = z} ⊂ R3 .
(b) W2 ⊂ R4 the subset of vectors with an even number of zero entries.
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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(c) W3 ⊂ F42 the subset of vectors with an even number of zero entries.
(d) W4 = {A ∈ M2,2 (C)|A = Ā} ⊂ M2,2 (C), where Ā is the matrix obtained by applying
complex conjugation to each entry of A.
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 4 (7 marks)
Consider the linear transformation T : P2 (R) → R3 defined by T (p) = (p(0), p(1), p(−1)) for
p = p(x) ∈ P2 (R).
(a) Find the matrix representation of T with respect to the standard bases {1, x, x2 } of
P2 (R) and {i, j, k} of R3 .
(b) Find a basis for the image of T .
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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(c) Find ordered bases B of P2 (R) and C of R3 such that [T ]C,B is the 3 × 3 identity matrix,
or prove that this is not possible.
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
Question 5 (4 marks)
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The Fibonacci numbers F0 , F1 , F2 , . . . are defined by
F0 = 0,
F1 = 1,
Fn = Fn−1 + Fn−2
(a) Use induction to prove that
n 1 1
Fn+1 Fn
=
1 0
Fn Fn−1
for all n ≥ 2.
for all n ≥ 1.
(b) Use the result of the previous part to prove that
(−1)n = Fn+1 Fn−1 − Fn2
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for all n ≥ 1.
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MAST10008 Accelerated Mathematics 1
Question 6 (11

5
Let A =  16
−12
marks)

0 2
1 8 .
0 −5
(a) Find the eigenvalues of A together with their multiplicities.
(b) Compute a basis for each of the eigenspaces of A.
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Semester 1, 2021
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
(c) Find an invertible matrix P and diagonal matrix D such that A = P DP −1 . (You do
not need to compute P −1 )
(d) Compute A728 .
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 7 (6 marks)
Suppose A is a 7 × 7 matrix such that A3 − A + I7 = O7 . (Here I7 is the 7 × 7 identity matrix,
and O7 is the 7 × 7 zero matrix.) Answer the following questions. Prove your assertions.
(Hint: Do not, under any circumstances, write out any 7 × 7 matrix!)
(a) Is 2 an eigenvalue of A?
(b) Does A have nullity zero?
(c) Is A invertible?
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 8 (7 marks)
Consider the real vector space P2 (R) of polynomials of degree at most 2. Define a function
hp, qi = p(0)q(0) + p(1)q(1) + p(2)q(2).
You may assume without proof that this is an inner product on P2 (R).
(a) Let U be the subspace of constant polynomials, and W = {a +bx +cx2 |3a+ 3b+ 5c = 0}.
Show that, for any p ∈ U and q ∈ W , p and q are orthogonal with respect to h , i.
(b) Apply the Gram–Schmidt process with respect to h , i to the standard basis {1, x, x2 } to
obtain an orthogonal basis (you do not have to normalise the vectors, although it may
help you to normalise the first two you compute).
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 9 (11 marks)
Give an example of each of the following, and prove that it is an example.
(a) An orthonormal (with respect to the dot product) basis {(u1 , u2 ), (v1 , v2 )} of R2 , such
that u1 u2 6= 0.
(b) A non-empty subset W ⊂ R3 that is closed under scalar multiplication, but is not a
subspace.
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
(c) A real vector space V , and a linear transformation T : V → V that is injective, but not
surjective.
(d) A square matrix A that cannot be written as a product of elementary matrices.
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 10 (9 marks)
Consider the function f : R2 → R given by
f (x, y) = 2x3 − 3x2 + 6y 2 − 12xy + 12x
(a) Find the direction of steepest descent on the graph of f at the point (0, 0, 0).
(b) Find the equation of the plane tangent to the graph of f at the point (0, 0, 0).
(c) Find and classify all the stationary points of f .
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 11 (5 marks)
Consider the definite integral
Z
π/2
I=
sin3 (2t) dt
−π/2
(a) Evaluate I using the complex exponential.
(b) Evaluate I using an alternative (much shorter) method.
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MAST10008 Accelerated Mathematics 1
Semester 1, 2021
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Question 12 (4 marks)
Let A be an invertible n × n real matrix and let pA (x) denote its characteristic polynomial.
Let pA−1 (x) denote the characteristic polynomial of the inverse matrix A−1 .
Show that
pA (x) = det(A)(−x)n pA−1 (x−1 )
(Hint: Consider det (A − xI)A−1 .
End of Exam — Total Available Marks = 87
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