The University of Sydney
MATH2022 Linear and Abstract Algebra
Semester 1 First Quiz Practice Exercises 2022
The First Quiz is at 11AM on March 24 (during lecture time), on canvas. This open book
quiz consists of eighteen multiple choice exercises, similar to the exercises below. Exactly one
alternative is correct for each question. You will have 50 minutes.
1. Which one of the following is not a field under addition and multiplication?
(a) Q (b) R (c) C (d) Z (e) Z13
2. If today is Thursday, what day of the week will it be after 20182018 days have elapsed?
(a) Friday
(d) Monday
(b) Saturday
(e) Tuesday
(c) Sunday
3. Which one of the following statements is true?
(a) 2
3
= 5 in Z11.
(d) 2
3
= 6 in Z13.
(b) 3
4
= 4 in Z13.
(e) 3
4
= 7 in Z11.
(c) 3
4
= 2 in Z7.
4. Consider the following matrix
M =
 1 3 23 4 3
1 1 1

with entries from Z7. Working over Z7, which of the following is true?
(a) detM = 0
(d) detM = 2
(b) detM = 4
(e) detM = 6
(c) detM = 5
5. Consider the following system of equations over Z5:
x + 2y + w = 1
2x + y + z = 2
x + y + 2z + 2w = 1
Working over Z5, how many distinct solutions are there for (x, y, z, w)?
(a) infinitely many
(d) exactly five
(b) no solutions
(e) exactly twenty-five
(c) exactly one
6. Find the unique solution to the following matrix equation
1 1 0 1
0 1 1 1
1 1 1 0
1 0 1 1


x
y
z
w
 =

0
1
0
1

working over Z2.
(a)

x
y
z
w
 =

1
1
0
1

(d)

x
y
z
w
 =

0
1
0
1

(b)

x
y
z
w
 =

0
0
1
1

(e)

x
y
z
w
 =

1
1
0
0

(c)

x
y
z
w
 =

0
1
1
1

7. Find the value of λ such that the system
x + z = 1
x + y + λz = 2
2x − λy − 4z = 6
is inconsistent over R, but has more than one solution over Z7.
(a) λ = 0
(d) λ = 3
(b) λ = 1
(e) λ = 4
(c) λ = 2
8. Consider the matrix
M =
 1 1 1 1 1 10 0 1 1 0 1
1 1 0 0 1 1

with entries from Z2. Which of the following is row equivalent to M and in reduced row
echelon form?
(a)
 1 1 0 0 0 10 0 1 1 0 0
0 0 0 0 1 0

(c)
 1 1 0 0 1 00 0 1 1 0 0
0 0 0 0 0 1

(e)
 1 1 0 0 0 00 0 1 1 1 0
0 0 0 0 0 1

(b)
 1 1 0 0 1 10 0 1 1 0 1
0 0 0 0 0 1

(d)
 1 1 0 0 0 00 0 1 1 0 0
0 0 0 0 1 1

9. Consider the following matrices over R, where θ is a real number:
Rθ =
[
cos θ − sin θ
sin θ cos θ
]
Tθ =
[
cos θ sin θ
sin θ − cos θ
]
Which one of the following statements is true?
(a) R3π/3 = I = T
2
π/2
(d) Rπ/2T2π/3Rπ/2 = T4π/3
(b) R32π/3 = I = T
3
2π/3
(e) Tπ/2R2π/3Tπ/2 = R4π/3
(c) R4π/4 = I = T
4
π/4
10. Consider the real matrix
M =
[
3 9
3 7
]
∼
[
1 3
3 7
]
∼
[
1 3
0 −2
]
∼
[
1 3
0 1
]
∼
[
1 0
0 1
]
and elementary matrices
E1 =
[
1 3
0 1
]
, E2 =
[
3 0
0 1
]
, E3 =
[
1 0
0 −2
]
, E4 =
[
1 0
3 1
]
.
Use the chain of equivalences above to find a correct expression for M as a product of
these elementary matrices.
(a) M = E2E4E1E3
(d) M = E2E1E3E4
(b) M = E2E4E3E1
(e) M = E3E1E4E2
(c) M = E4E3E1E2
11. Suppose that a, b, c, x are elements of a group G such that
axcba−1 = b .
Which one of the following is a correct expression for x ?
(a) x = a−1ba(cb)−1
(d) x = c−1(ba−1)−1a−1b
(b) x = (ba)−1c−1ba
(e) x = (ba−1)−1a−1bc−1
(c) x = a−1b(cb)−1a
12. Consider the permutations
α = (5 2 1 4 3) , β = (1 3)(2 4 6) , γ = (1 2 4)(3 5 6)
of {1, 2, 3, 4, 5, 6} expressed in cycle notation. Which one of the following is correct?
(a) α and γ are odd, and β is even.
(c) α and β are even, and γ is odd.
(e) β and γ are even, and α is odd.
(b) α and γ are even, and β is odd.
(d) α and β are odd, and γ is even.
13. Consider the group G of symmetries of a square, generated by a rotation α and a reflec-
tion β. Simplify the following expression in G:
αβαβ3α−3β−1α−1 =
(a) αβ
(d) α3
(b) α2β
(e) α2
(c) α3β
14. Consider the permutations
α = (1 2 3 4)(5 6 7) , β = (1 3)(2 4) , γ = (1 2 3)(4 5)(6 7)
of {1, 2, 3, 4, 5, 6, 7} expressed in cycle notation. Simplify the permutation
δ = αβγ−1 ,
composing from left to right:
(a) δ = (1 5 7 4 2 3)
(c) δ = (1 4 7 5)
(e) δ = (1 5 7 4)
(b) δ = (1 5 7 4)(3 2)
(d) δ = (1 3 2 4 7 5)
15. Consider the permutations
α = (1 3)(2 4 6 5) and β = (1 4 2 5)(6 3)
of {1, 2, 3, 4, 5, 6} expressed in cycle notation. Which one of the following is a correct
expression for the permutation
γ = β−1αβ
where we compose from left to right?
(a) γ = (4 6)(5 2 1 3)
(d) γ = (4 6)(5 3 1 2)
(b) γ = (5 6)(4 3 1 2)
(e) γ = (5 6)(4 1 3 2)
(c) γ = (4 6)(1 5 2 3)
16. Which one of the following configurations is possible to reach from the 8-puzzle
1 2 3
4 5 6
7 8
by moving squares in and out of the space?
(a)
4 1 5
6 2 3
7 8
(d)
2 3 4
8 7 5
1 6
(b)
7 6 4
2 8 3
1 5
(e)
4 8 2
5 7 1
3 6
(c)
8 6 4
3 1 5
2 7