(a) Friday (b) Saturday (d) Monday (e) Tuesday
4. Which one of the following statements is true?
(c) Sunday
(c) 34 =2inZ7.
(a) 23 =5inZ11. (d) 32 =6inZ13.
5. Consider the following matrix
(b) 43 =4inZ13. (e) 43 =7inZ11.
132 M=343
The University of Sydney
MATH2022 Linear and Abstract Algebra
Semester 1 First Quiz Practice Exercises 2023
The first quiz is on March 23 on canvas. It will be an online quiz of twelve multiple choice questions for which you will have 40 minutes. These questions will be similar to the ones 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 under addition and multiplication?
(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?
111
with entries from Z7. Working over Z7, which of the following is true?
(a) detM =0 (b) detM =4 (d) detM=2 (e) detM=6
6. Consider the following system of equations over Z5:
x + 2y + w = 1 2x + y + z = 2 x+ y+2z+2w=1
(c) detM =5
Working over Z5, how many distinct solutions are there for (x, y, z, w)?
(a) infinitely many (b) no solutions (c) exactly one (d) exactly five (e) exactly twenty-five
7. Find the unique solution to the following matrix equation 1 1 0 1x 0
working over Z2. x 1
0 1 1 1y=1 1 1 1 0 z 0
1011w1 x 0
x 0
(a) y = 1 z 0
(c) y = 1 z 1
x 0 (d) y = 1
x 1 (e) y = 1
z 0
w1 w0
z 0 8. 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
9. Consider the matrix
(b) λ=1 (e) λ=4
111111 M=001101
110011
(c) λ=2
y = 0
z 1
(b) w1w1w1
with entries from Z2. Which of the following is row equivalent to M and in reduced row echelon form?
110001 110011 (a) 001100 (b) 0 0 1 1 0 1
000010 000001
110010 (c) 001100
000001
110000 (e) 0 0 1 1 1 0
000001
110000 (d) 0 0 1 1 0 0
000011
10. Consider the following matrices over R, where θ is a real number:
? cosθ −sinθ ? ? cosθ sinθ ?
Rθ = sinθ cosθ Tθ = sinθ −cosθ Which one of the following statements is true?
(a) R3 = I = T2 π/3 π/2
(d) Rπ/2T2π/3Rπ/2 = T4π/3 11. Consider the real matrix
(b) R3 = I = T3 (c) 2π/3 2π/3
(e) Tπ/2R2π/3Tπ/2 = R4π/3
R4 = I = T4
?39? ?13? ?1 3? ?13? ?10? M= 37 ∼ 37 ∼ 0−2 ∼ 01 ∼ 01
and elementary matrices
?13? ?30? ?1 0? ?10?
E1= 01 ,E2= 01 ,E3= 0−2 ,E4= 31 .
Use the chain of equivalences above to find a correct expression for M as a product of
these elementary matrices.
(a) M = E2E4E1E3 (b) M = E2E4E3E1 (c) M = E4E3E1E2
(d) M = E2E1E3E4 (e) M = E3E1E4E2
12. Suppose that A, B and P are real square matrices such that P is invertible and λ ∈ R
such that
where I denotes the identity matrix. Which of the of the following is a correct expression
for A ?
(a) A=P−1BP +λI (b) A=PBP−1 +λI
(d) A=P−1B+λP−1 (e) A=P−1BP +λP−1 13. 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 ?
(c) A=P−1BP +λP
(c) x = a−1b(cb)−1a
(a) x = a−1ba(cb)−1
(d) x = c−1(ba−1)−1a−1b
14. Consider the permutations
α = (52143),
(b) x = (ba)−1c−1ba
(e) x = (ba−1)−1a−1bc−1
P(A−λI)P−1 =B,
β = (13)(246),
of {1, 2, 3, 4, 5, 6} expressed in cycle notation. Which one of the following is correct?
(a) αandγareodd,andβiseven. (b) αandγareeven,andβisodd. (c) αandβareeven,andγisodd. (d) αandβareodd,andγiseven. (e) βandγareeven,andαisodd.
γ = (124)(356)
π/4
π/4
15. 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 (b) α2β
(d) α3 (e) α2 16. Consider the permutations
α = (1234)(567), β = (13)(24),
of {1, 2, 3, 4, 5, 6, 7} expressed in cycle notation. Simplify the permutation
δ = αβγ−1 ,
(a) δ=(157423) (e) δ=(1574) (d) δ=(132475)
(c) δ=(1475) (b) δ=(1574)(32) 17. Consider the permutations
α = (13)(2465) and β = (1425)(63)
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=(46)(5213) (b) γ=(56)(4312) (c) γ=(46)(1523) (d) γ=(46)(5312) (e) γ=(56)(4132)
18. Consider the permutations
α = (132)(465) and γ = (425)(613)
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=(146)(235) (b) β=(142)(365) (c) β=(1623) (d) β=(1364)(35) (e) β=(1326)
composing from left to right:
(c) α3β
γ = (123)(45)(67)
19. Which one of the following configurations is possible to reach from the 8-puzzle
123 456 78
by moving squares in and out of the space?
415 764 864 (a) 623 (b) 283 (c) 315
781527
234 482 (d) 875 (e) 571
16 36
20. Which one of the following configurations is impossible to reach from the 8-puzzle
123 456 78
by moving squares in and out of the space?
24 123 (a) 815 (b) 68 763 754
14 314 (d) 327 (e) 82 658 657
712 (c) 36 485
