# 代写接单

(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 1x 0

working over Z2. x 1

0 1 1 1y=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

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