MATH 2022 Week03Worksheet

MATH 2022 Week 3 worksheet

QY Putm=

[55-56]

.

Over R,

#det M = -and

M' '

=

Over 217 ,

m=EEI=EiT,

det M= .and

M"

=

Why does M' ' not exist over ZL

, ,?

- ]

QY Solve

x + Z = O

y + z = o

x + y= 1

over R :

k::L:t

over 273 by converting previous solution :

-Why is there no solution over Zz ?

.

QY Solve the following homogeneous system

over Zz !

x,

t nz t Nz + ks = 0

a,+ nz it notas = o

se,

+ ×3 + ks = 0

se,+ xz try

= 0

after first row reducing the following matrix

to reduced row echelon form :

f :::::

i ::::D .

QYI Consider the system

n + y = 1

n + z =

7+1n +

XZ= 0

@1 Row reduce to row echelon form :

[ :: :p't'tI o X o

(b) Over R,for which X is the system

Ci ) inconsistent ? M'

-

? 17di ) having a unique solution

The solution becomes-

(c) What happens over 723 ?

(d) What happens over Zz ?

Q5/ Put x = ( i 2345 ) ( 678 910 ).

find 22 =

÷4=

5

L =

and complete the multiplication table :

1 L ×' £3 Lt

1

:÷L

How does this comparewith the addition

table for 125 ?

-The group G=k > is cyclic ? True False

Q6/Put L = C 123 )

, p= ( 45 )

,F- xp = (123×45)

find

z2=83 =

84 =

*6 =

and complete the multiplication table :

1 8 82 y'

y4

y5

1

8

:'y5

How does this comparewith the addition

table for 26 ?

-The group G=k

, p >is cyclic ? True False

Q7/Put L= C i 2) ( 34 )

, p = ( 1 3) C 24 ),

8 = Lp = ( 1 2) ( 3 4) ( 1 3) ( 24 ) =

Find x'

=p'

= 82 =

pr =

jx =

and complete the multiplication table :

1 × p 8

1

:8

How does this comparewith the composition

table for symmetries of the rectangle ?

-The group G=k

, p >is cyclic ? True False

Q8/ Consider polynomials with coefficients

from Zz={ oil }.

Put

F = { 0,1 ,r

,Ita }

.

Complete the addition table for f :

+ 0 I a Hk

0

:ltn

Complete the multiplication table for F if

(a) It ih=o 0 I a Ita

:

: aor

Cb ) ltntxhio0 I a Ita

:

: aWhich of (a) or (b) yields a field ? D