The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH2022: Linear and Abstract Algebra Semester 1, 2023
Lecturer: Alexander Sherman and Pieter Roffelsen
Due 11:59pm Sunday, May 14 2023.
This assignment contains three questions and is worth
5% of your total mark. It should uploaded using Turnitin
through the MATH2022 Canvas portal. Please include
your SID but not your name, as anonymous marking will
be implemented.
1. Suppose that in 2023, 90% of the Australian population eats meat, 6% is vegetarian, and
4% is vegan. Suppose further that each year the following evolution occurs:
1. 90% of meat eaters remain as such, while 5% become vegetarian and 5% become
vegan;
2. 80% of vegetarians remain as such, while 10% start eating meat and 10% become
vegan; and
3. 60% of vegans remain as such, while 30% become vegetarian and 10% start eating
meat.
(a) Write a stochastic matrixM that describes the evolution of the proportion of meat
eaters, vegetarians, and vegans in Australia.
(b) What percentage of Australians can we expect to be vegan in 2024?
(c) Find the steady state probability vector of M .
(d) If the above stated trends continue, what percentage of Australians should we
expect to be either vegan or vegetarian in 2075, to a good approximation?
2. For each of the following statements, indicate whether they are true or false. If a statement
is true, prove it. If a statement is false, give a counterexample.
(a) Every abelian group is cyclic.
(b) There is no 10× 7 matrix of rank 8.
(c) Every matrix of nullity 0 is invertible.
(d) Let p be a prime number. Then Zp has p−1 generators as a group under addition.
3. Throughout this exercise, i =
√−1 ∈ C. Recall that if z = a + bi ∈ C is a complex
number where a, b ∈ R, then z = a − bi ∈ C is called the complex conjugate of z. You
may freely use the following facts about complex conjugation for any w, z ∈ C:
z + w = z + w, zw = z · w,
and that z = z if and only if z ∈ R. Consider the following set of complex 2× 2 matrices:
H =
{[
z −w
w z
]
: w, z ∈ C
}
.
Copyright© 2023 The University of Sydney 1
Finally, put B = {1, i, j,k}, where
1 =
[
1 0
0 1
]
, i =
[
i 0
0 −i
]
, j =
[
0 i
i 0
]
, k =
[
0 −1
1 0
]
.
(a) Verify that B is a subset of H and that the following equations holds:
i2 = j2 = k2 = ijk = −1.
(b) Show that H is closed under matrix addition and scalar multiplication by real
numbers. Is it closed under scalar multiplication by complex numbers?
(In particular, this shows H is a subspace of the vector space of 2 × 2 complex
matrices viewed as a real vector space.)
(c) Prove that B is a basis for H, regarded as a real vector space. What is the dimension
of H?
(d) Show that H is closed under matrix multiplication.
(e) Check that if A ∈ H and A is not the zero matrix, then A is invertible and A−1 ∈ H.
(f) Does H form a field under the operations of matrix addition and matrix multipli-
cation? Give reasoning for your answer.
(In fact, H is a famous arithmetic discovered by William Rowan Hamilton, known as
the quaternions. The quaternions have important applications in computer graphics and
other fields. The equations in (a) became famous after he carved them in stone on Broom
Bridge in Dublin in 1843.)