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Assignment: Determinants Date: 17thDecember, 2009 Batch: Target

1. Without expanding evaluate the determinant

(i)352

!!

51"1

(ii)bac

acb

cba

+

+

+

1

1

1

(iii)baabba

accaac

cbbccb

+

+

+

22

22

22

(iv)2

2

2

1

1

1

cc

bb

aa

2. #rove that ))()()((222

++=

+++

3. $valuate

321

321

321

CCC

CCC

CCC

zzz

yyy

xxx

". %sing properties o& determinants' sho that )

1

1

1

2

2

2

=

abcc

cabb

bcaa

5. #rove that3

222

222

222

)(2

)(

)(

)(

cbaabc

bacc

bacb

aacb

++=

+

+

+

*. +ho that ))(( 222 cbacba

cabba

acbca

bccba

++++=

+

+

+

!. ,et

2

)13(23

12

2

)1(

1

2=

+

==

n

rrr thatShow

nnzr

nyr

nnxr

-. ,et the three digit numers CandBA *23'2- ' here A' B' / are integers eteen and ' e divisile

0 a &ixed integer . +ho that the determinant22

-

*3

B

C

A

is also divisile 0 the same integer .

. ,et

333)1(

2"2)1(

*1

1233

22=

==

n

aaa thatShow

nnna

nna

na

1. or a &ixed positive integer n' i&)"()3()2(

)3()2()1(

)2()1(

+++

+++

++

=

nnn

nnn

nnn

D then sho that ")( 3

n

Dis divisile 0

n

11. Without expanding' prove that BxA

xxxx

xxxx

xxxx

+=

++

+

++

121232

333132

21

2

2

2

' here A and B are determinants o&

343 suare matrices not involving x.

12. $xpress222

222

222

)()()(

)()()(

)()()(

zcycxc

zbybxb

zayaxa

= as the product o& to determinants and evaluate it.

13. $xpress the determinant

1)cos()cos(

)cos(1)cos(

)cos()cos(1

= as the product o& to determinants and

hence sho that )=