The University of Sydney
MATH2022 Linear and Abstract Algebra
Semester 1 First Quiz Practice Exercises 2021
The First Quiz (Take Home) is due before midnight on Friday 2 April 2021 and consists of
fifteen multiple choice exercises (together with written justifications), similar to a selection
from the exercises below.
1. Which one of the following forms a field under addition and multiplication?
(a) N (b) Z2 (c) Z4 (d) Z6 (e) Z8
2. Which one of the following is not a field?
(a) Q (b) R (c) C (d) Z (e) Z13
3. 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
4. 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.
5. 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
6. 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
7. 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

8. Find the value of λ such that the system
x + 2z = 2
−x + λy + z = −1
x − y − λz = 1
is inconsistent over Z5, but has a unique solution over R.
(a) λ = 0
(d) λ = 3
(b) λ = 1
(e) λ = 4
(c) λ = 2
9. 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

10. 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) R3pi/3 = I = T
2
pi/2
(d) Rpi/2T2pi/3Rpi/2 = T4pi/3
(b) R32pi/3 = I = T
3
2pi/3
(e) Tpi/2R2pi/3Tpi/2 = R4pi/3
(c) R4pi/4 = I = T
4
pi/4
11. Consider the real matrix
M =
[
0 2
3 9
]
∼
[
3 9
0 2
]
∼
[
1 3
0 2
]
∼
[
1 3
0 1
]
∼
[
1 0
0 1
]
and elementary matrices
E1 =
[
1 0
0 2
]
, E2 =
[
3 0
0 1
]
, E3 =
[
1 3
0 1
]
, E4 =
[
0 1
1 0
]
.
Use the chain of equivalences above, or otherwise, to find a correct expression for M as a
product of these elementary matrices.
(a) M = E4E2E3E1
(d) M = E4E3E2E1
(b) M = E4E2E1E3
(e) M = E2E1E3E4
(c) M = E3E2E1E4
12. Suppose that A, B and P are real square matrices such that P is invertible and λ ∈ R
such that
P (A− λI)P−1 = B ,
where I denotes the identity matrix. Which of the of the following is a correct expression
for A ?
(a) A = P−1BP + λI
(d) A = P−1B + λP−1
(b) A = PBP−1 + λI
(e) A = P−1BP + λP−1
(c) A = P−1BP + λP
13. Suppose that a, b, c, d, g are elements of a group G such that
abgc−1 = d .
Which one of the following is a correct expression for g ?
(a) g = a−1b−1dc
(d) g = b−1a−1dc
(b) g = dca−1b−1
(e) g = c(ab)−1d
(c) g = (ab)−1cd
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. Which one of the following is correct?
(a) β and γ are even, and α is odd.
(c) α and γ are even, and β is odd.
(e) α and γ are odd, and β is even.
(b) α and β are even, and γ is odd.
(d) α and β are odd, and γ is even.
15. Consider the permutation
α = (1 2 3)(6 3 2)(5 4 6 3 2)
of {1, 2, 3, 4, 5, 6} expressed in cycle notation where we compose from left to right. Which
one of the following is a correct equivalent expression?
(a) α = (1 2 3)(5 6 4)
(d) α = (1 4)(3 6 5 2)
(b) α = (1 3)(2 5 4 6)
(e) α = (1 4 2)(3 5 6)
(c) α = (1 3)(2 6 4 5)
16. 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)
17. 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)
18. Consider the permutations
α = (1 3)(4 2 5 6) and γ = (4 5)(1 3 2 6)
of {1, 2, 3, 4, 5, 6} expressed in cycle notation. Which one of the following is a correct
expression for a permutation β with the property
γ = β−1αβ
where we compose from left to right?
(a) β = (1 6)(2 3)
(d) β = (1 4)(2 6)(3 5)
(b) β = (1 4 2 3 6 5)
(e) β = (1 4 2 6 3 5)
(c) β = (1 4 6)(2 3 5)
19. 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
20. Which one of the following configurations is impossible to reach from the 8-puzzle
1 2 3
4 5 6
7 8
by moving squares in and out of the space?
(a)
2 4
8 1 5
3 6 7
(d)
8 3 4
2 7
6 5 1
(b)
7 2 4
6 8
1 5 3
(e)
7 1
6 4 2
3 5 8
(c)
4 8 2
3 6
1 7 5

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