The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH2022: Linear and Abstract Algebra Semester 1, 2024
Lecturer: Leah Neves and Sam Jeralds
Due 11:59pm Sunday 14 April 2024.
This assignment contains four questions and is worth 5% of your total
mark. It must be uploaded through the MATH2022 Canvas page
https://canvas.sydney.edu.au/courses/56888/assignments/518805.
Please include your SID but not your name, as anonymous marking will be
implemented.
1. Let G be a finite group with an even number of elements. Show that there must be some
element α in G, α 6= e (e the identity element) such that α2 = e.
2. Consider the matrix
A =
1 3 11 1 2
0 2 4

over Z5.
(a) Is A invertible over Z5? Justify your answer.
(b) How many solutions does the system of equations
x+ 3y + z = 2
x+ y + 2z = 3
2y + 4z = 4
have over Z5?
3. 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) For two permutations α, β in the symmetric group Sym(n), if α is odd, then the
conjugation β−1αβ is also odd.
(b) For three square matrices A, B, and C of the same size, if AC = BC, then A = B.
(c) Let A be a 3× 3 matrix over R such that AT = −A (where AT is the transpose).
Then det(A) = 0.
4. Let A, B be two square matrices of the same size over a field F such that AB = BA. If
~v is an eigenvector of A, show that the product (B · ~v) is also an eigenvector of A with
the same eigenvalue as ~v.
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