MATH 2022 Week04Worksheet

MATH 2022 Week 4 Worksheet

Working over 223 , puta "

m=l÷'It.

find

detm.lk's? I =

Complete the following row reduction

to find M' '

:

KIM ::D .

Q2/Working over 23 we have

at :L;] allow # E.⇒at:!:D at :::d=± .

Find matrices that producethese

row equivalences by premuttiplication :

(a) A = (b) B =

(c) C = (d) D=

Express I as a product of M

and A ,B , C ,D in some order :

I =

Now find an expression for M^

in terms of A,B,

c,D .

M =

( This yields an LDU - decomposition . )

QY Put m=[ },;] and

L=['

a ;] ,

D= [ !:] ,u=[ 'of ]

where a,b,

a,d ¥ 0

.

find

i'

=L ] ,

is'

= [ ] ,

n''

=L ].

Row reduce M to I in three main steps :

m= L ! ?t]~[ !-9 , ]~[

]~['oI=I.

Use these steps to write I as a product

of M and matrices of the form L,D,U '

.

I :[][ ][ ]m .

Now write M as a product of matrices

of the form L,D

,n :

M =

( You will have found the LDU - decompositionof M

. )

Q4/ Consider the rotation matrix

Rite( ¥ E ]Tr Eu Fu

and the reflection matrixTit= [ 9! ]

I

Multiply out and simplify :

TithRtq Ttg =

Rita Ttyz RTL,=

What geometric propertiescorrespond to

these calculations ?

QSY I •i

• 2

÷;!iIi÷;÷i!IE Iii:c:bComplete the following & interpret geometrically :

L = ( 1234 ) 1/4 rotation

B = C 1 27 ( 34 ) reflection in vertical axis is

one+=p

2= .

xp = l

:tip =

R alt

x3p =

Complete :

ftp. ,flats =

flip = ,I 'px=

px=

;xpa =

Simplify (avoiding cycles) :

patp3i2p→i' p =

QY'

•III'÷i÷f¥÷f.fi: a=an⇒

•

• p = ( 25 ) ( 34 )

Complete the following l interpret geometrically :

2 = ( 12345 ) Ys rotation

p = ( 25 ) ( 34 ) reflection in vertical axis

÷3=

÷i=p2=

xp =

seen

4p=

flap =,Ilpx

-pa= , xpx=

( xp )' ' at ( xp) =

, p xp =

Q7/ True or false ?

(a) ( 1 2) ( 347156 ) Is even.

T f

(b) ( 123 ) is odd.

T F

q ( 123 ) = ( 12 ) ( 23 ) T f

(d) ( 123 ) = ( 1 3) ( 23 ) Tf

(e) ( 321 ) = ( 1 2) ( 23 ) Tf

(f) ( 1234 ) is even.

Tf

(g) ( 136 ) ( 479 )( 258 ) is odd.

TF

( h ) If L= ( 134 ) ( 25 ), p=( 263 ) ,

8=( 34)

theni ) ptxp = ( 12 4) ( 65 ) T f

di ) I 'px= ( 564 ) TFCiiy 8ps = ( 426 )

T F

C ) 8xV= ( 143 ) ( 25 )

TFe) I

'8£=(44 =p

'

'8p T F

Q8/ The starting configuration of the

8- puzzle looks like 1 23

4 5 6

78

True or false ?

(a) ' 23 is possible T F4 6 8

7 52I 3

(b)456

is possible T F

78

(c)2 ' 3

is impossible T F6 458

7

(d)2 1 4

is impossible T F7 3

856