8/11/2019 3.3 Exercise

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MATRIX AND DETERMINANTS

1. If a matrix has 28 elements, what are the possible orders it can have? What if it

has 13 elements?

2. In the matrix

=

5

250

32

1

2 yx

xa

A , write

!i" #he order of the matrixA

!ii" #he n$mber of elements

!iii" Write elements 123123 ,, aaa

3. %onstr$ct 22a matrix where

!i" ( )

2

2 2ji

aij

=

!ii" jiaij 32 +=

&. %onstr$ct a !3 2" matrix whose elements are 'iven b( jxea xi

ij sin.

= .

5. )ind the val$es of aand bif BA = , where

+=

*8

3& baA ,

++=

bb

baB

58

222

2

2

8/11/2019 3.3 Exercise

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*. If possible, find the s$m of the matricesAandB, where

=

32

13A , and

=

*ba

zyxB .

+. If

=

325

113X and

=

&2+

112Y , find

!i" YX + !ii" YX 32

!iii" matrixZs$ch that ZYX ++ is a -ero matrix.

8/11/2019 3.3 Exercise

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8. )ind non-ero val$es ofxsatisf(in' the matrix e/$ation

( )( )

+=

+

x

x

x

x

x

xx

*10

2&82

&&

582

3

22 2.

. If

= 11

10A and

=

01

10B , show that ( ) ( )

22 BABABA + .

8/11/2019 3.3 Exercise

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10. )ind the val$e ofxif [ ] Ox

x =

2

1

2315

152

231

11 .

11. how that

=

21

35A satisfies the e/$ation OIAA = +32 and hence find 1A .

12. )ind the matrix satisf(in' the matrix e/$ation

8/11/2019 3.3 Exercise

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=

10

01

35

23

23

12A .

13. )indA, if

=

3*3

121

&8&

3

1

&

A .

8/11/2019 3.3 Exercise

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1&. If

=

02

11

&3

A and

=

&21

212

B, then verif( ( ) 222 ABBA .

15. If possible, findBAandAB, where

==21

32

1&

,&21

212BA .

1*. how b( an example that for OABOBOA = ,, .

1+. iven

=

*3

0&2A and

=

31

82

&1

B . Is ( ) ABAB = ?

18. olve forxandy

8/11/2019 3.3 Exercise

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0

11

8

5

3

1

2 =

+

+

yx .

1. IfXand Yare 22 matrices, then solve the followin' matrix e/$ations forXand Y

=+

=+

51

2223,

0&

3232 YXYX .

20. If [ ]53=A , [ ]3+=B , then find a non-ero matrix Cs$ch that BCAC= .

21. ive an example of matricesA, Band Cs$ch that ACAB = , whereAis non-ero

matrix, b$t CB .

8/11/2019 3.3 Exercise

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22. If

=

= &3

32,

12

21BA and

=0101C , verif(

!i" ( ) ( )BCACAB = !ii" ( ) ACABCBA +=+ .

23. If

=

z

y

x

P

00

00

00

and

=

c

b

a

Q

00

00

00

, prove that

QP

zc

ybax

PQ =

=

00

00

00.

8/11/2019 3.3 Exercise

9/47

2&. If [ ]312 A=

1

0

1

110

011

101

, findA.

25. If [ ]

==

*+8

&35,12 BA and

=

201

121C , verif( that ( ) ( )ACABCBA +=+ .

2*. If

=

110

312

101

A , then verif( that ( )IAAAA +=+2 , whereIis 3 3 $nit matrix.

2+. If

=

&3&

210A and

=

*2

31

0&

B , then verif( that

8/11/2019 3.3 Exercise

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!i" ( ) AA = !ii" ( ) ABAB =

!iii" ( ) ( )AkkA = .

28. If

=

=

3+

&*

21

,

*5

1&

21

BA , then verif( that

!i" ( ) BABA +=+ 22

!ii" ( ) BABA = .

2. how that AA and AA are both s(mmetric matrices for an( matrixA.

30. et A and B be s/$are matrices of the order 3 3. Is ( ) 222

BAAB = ? iven

reasons.

31. how that if A and B are s/$are matrices s$ch that BAAB = , then

( ) 222 2 BABABA ++=+ .

8/11/2019 3.3 Exercise

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32. et

=

=

=

21

02,

51

0&,

31

21 CBA and 2,& == ba . how that

!i" ( ) ( ) CBACBA ++=++

!ii" ( )CABBCA ="!

!iii" ( ) bBaBBba +=+

!iv" ( ) aAaCACa =

!v" ( ) AA TT

=

!vi" ( ) TTT

ABAB =

!vii" ( ) BCACCBA =

!viii" ( ) TTT BABA =

33. If

=

cossin

sincos

A, then show that

=

2cos2sin

2sin2cos2

A.

8/11/2019 3.3 Exercise

12/47

3&. If

=

= 01

10,

0

0B

x

x

A and 12

=x , then show that ( ) 222

BABA +=+ .

35. 4erif( that IA =2 when

=

&33

&3&

110

A .

3*. rove b( 6athematical Ind$ction that ( ) ( )= nn AA , where Nn for an( s/$are

matrixA.

3+. )ind inverse, b( elementar( row operations !if possible", of the followin'

matrices

!i"

+5

31!ii"

*2

31.

8/11/2019 3.3 Exercise

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38. If

=

++ *0

8

*

& w

yxz

xy, then find val$es ofx, y, zand w.

3. If

=

12+

51A and

=

8+

1B , find a matrix Cs$ch that CBA 253 ++ is a n$ll matrix.

&0. If

= 2&

53

A, then find IAA 1&5

2

. 7ence, obtain3

A .

8/11/2019 3.3 Exercise

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&1. )ind the val$es of a, b, c andd,if,

+

++

=

3

&

21

*3

dc

ba

d

a

dc

ba .

&2. )ind the matrixAs$ch that

=

1522

521

1081

&3

01

12

A .

&3. If

=

1&

21A , find IAA +22 ++ .

8/11/2019 3.3 Exercise

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&&. If

=

cossin

sincosA , and AA =1 , find val$e of .

&5. If the matrix

01

12

30

c

b

a

is a sew s(mmetric matrix, find the val$es of a, b and

c.

&*. If ( )

=

xx

xxxP

cossin

sincos, then show that ( ) ( ) ( ) ( ) ( )xPyPyxPyPxP .. =+= .

&+. IfAis s/$are matrix s$ch that AA =2 , show that ( ) IAAI +=+ +3 .

&8. IfA, Bare s/$are matrices of same order andBis a sews(mmetric matrix, show

that BAA is sew s(mmetric.

&. If BAAB = for an( two s/$are matrices, prove b( mathematical ind$ction that

( ) nnn BAAB = .

50. )ind zyx ,, if

=

zyx

zyxzy

A20

satisfies 1= AA .

8/11/2019 3.3 Exercise

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51. If possible, $sin' elementar( row transformations, find the inverse of the

followin' matrices

!i"

323

135

312

!ii"

111

221

332

!iii"

310

015

102

52. 9xpress the matrix

21&

211

132

as the s$m of a s(mmetric and a sew s(mmetric

matrix.

53. %onstr$ct a matrix 22= ijaA whose elements ija are 'iven b(

jxea ixij sin2

= .

8/11/2019 3.3 Exercise

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5&. If

====

0+5

8*&,2

1,13&

231,21

32DCBA , then which of the s$ms DCCBBA +++ ,, and DB + is defined?

55. how that a matrix which is both s(mmetric and sew s(mmetric is a -ero matrix.

5*. If [ ] Oxx =

803

2132 , find the val$e ofx.

8/11/2019 3.3 Exercise

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5+. IfAis 3 3 invertible matrix, then show that for an( scalar k !non-ero", kA is

invertible and ( ) 11 1

= Ak

kA .

58. 9xpress the matrix Aas the s$m of a s(mmetric and a sew s(mmetric matrix,

where

=

&21

53+

*&2

A .

5. If

=

321

102

231

A , then show thatAsatisfies the e/$ation 0113& 23 =+ IAAA .

*0. et

=

21

32A . #hen show that 0+&2 =+ IAA . :sin' this res$lt calc$late 5A

also.

9val$ate, $sin' the properties of determinants

8/11/2019 3.3 Exercise

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*1.

11

112

++

+

xx

xxx*2.

zayx

zyax

zyxa

+

+

+

*3.

0

0

0

22

22

22

zyzx

yzyx

xzxy

*&.

zzyzx

yzyyx

zxyxx

3

3

3

++

*5.

&

&

&

+

+

+

xxx

xxx

xxx

**.

baccc

bacbb

aacba

22

22

22

:sin' the properties of determinants, prove that

*+. 0

22

22

22

=

+

+

+

yxxyyx

xzzxxz

zyyzzy

*8. xyz

yxxy

xxzz

yzzy

&=

+

+

+

8/11/2019 3.3 Exercise

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*. ( ) 3

2

1

133

1212

1122

=++

++

aaa

aaa

+0. If 0=++ CBA , then prove that 0

1coscos

cos1cos

coscos1

=

AB

AC

BC

.

+1. If the coordinates of the vertices of an e/$ilateral trian'le with sides of len'th ;aone of these

122. IfAandBare invertible matrices, then which of the followin' is not correct?

!a" 1.

= AAAadj !b" ( ) ( )[ ] 11 detdet = AA

!c" ( ) 111 = ABAB !d" ( ) 111 +=+ ABBA

123. If zyx ,, are all different from -ero and 0

111

111

111

=

+

+

+

z

y

x

, then val$e of

111 ++ zyx is

!a" zyx !b" 111 zyx

!c" zyx !d" 1

12&. #he val$e of the determinant

xyxyxyxxyx

yxyxx

22

2

++++

++

is

!a" ( )yxx +2 !b" ( )yxy +2

!c" ( )yxy +23 !d" ( )yxx +2+

8/11/2019 3.3 Exercise

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125. #here are two val$es of awhich maes determinant, 8*

2&0

12

521

=

=

a

a , then s$m of

these n$mber is

!a" & !b" 5 !c" & !d"

12*. et

1

1

1

2

2

2

zCz

yBy

xAx

=and

xyzxzyzyx

CBA

=1, then

!a" =1 !b" 1

!c" 01 = !d" >one of these

12+. If !yx , , then the determinant

( ) ( ) 1sincos1cossin1sincos

yxyx

xxxx

++

= lies in the interval

!a" [ ]2,2 !b" [ ]1,1

!c" [ ]1,2 !d" [ ]2,1

Fill in the blanks

128. AAAAAAAAAAA matrix is both s(mmetric and sew s(mmetric matrix.

12. $m of two sew s(mmetric matrices is alwa(s AAAAAAAAAAA matrix.

130. #he ne'ative of a matrix is obtained b( m$ltipl(in' it b( AAAAAAAAAAAAAA.

131. #he prod$ct of an( matrix b( the scalar AAAAAAAA is the n$ll matrix.

8/11/2019 3.3 Exercise

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132. matrix which is not a s/$are matrix is called a AAAAAAAAAAA matrix.

133. 6atrix m$ltiplication is AAAAAAAAAAAAAAAA over addition.

13&. IfAis a s(mmetric matrix, then 3A is a AAAAAAAAAAAA matrix.

135. IfAis a sew s(mmetric matrix, then 2A is a AAAAAAAAAAAAAAA.

13*. IfAandBare s/$are matrices of the same order, then

!i" ( )AB B AAAAAAAAAAAAA.

!ii" ( )kA B AAAAAAAAAAAAA. !kis an( scalar"

!iii" ( )[ ] BAk B AAAAAAAAAAA.

13+. IfAis sew s(mmetric, then kA is a AAAAAAAAAAAAA. !kis scalar"

138. IfAandBare s(mmetric matrices, then

!i" BAAB is a AAAAAAAAAAAAA.

!ii" ABBA 2 is a AAAAAAAAAAAAA.

13. IfAis s(mmetric matrix, then ABB is AAAAAAAAAAAAAA.

1&0. If AandBare s(mmetric matrices of same order, then AB is s(mmetric if and

onl( if AAAAAAAAAAAAAA.

1&1. In appl(in' one or more row operations while findin' 1A b( elementar( row

operations, we obtain all -eros in one or more, then 1A AAAAAAAAAAAAAA.

1&2. IfAandBare two sew s(mmetric matrices of same order, then ABis s(mmetric

matrix is AAAAAAAAAAA.

1&3. IfAandBare matrices of same order, then ( ) BA 23 is e/$al to AAAAAAAAAA.

1&&. ddition of matrices is defined if order of the matrices is AAAAAAAAAAAAA.

1&5. IfAis a matrix of order 3 3, then A3 B AAAAAAAAAAAAAAA.

1&*. IfA is invertible matrix of order 3 3, then1

A AAAAAAAAAAAAAAAAAA.

8/11/2019 3.3 Exercise

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1&+. If !zyx ,, , then the val$e of determinant

( ) ( )( ) ( )( ) ( ) 1&&&&

13333

12222

22

22

22

xxxx

xxxx

xxxx

+

+

+

is

e/$al to AAAAAAAAAAAAAA.

1&8. If 02cos = , then

2

cos0sin

0sincos

sincos0

B AAAAAAAAAAAAA.

1&. IfAis a matrix of order 3 3, then ( ) 12 A B AAAAAAAAAAAAAA.

150. If Ais a matrix of order 3 3, then n$mber of minors in determinant of A are

AAAAAAAAAAAA.

151. #he s$m of the prod$cts of elements of an( row with the cofactors of

correspondin' elements is e/$al to AAAAAAAAAAAAAAA.

152. If =x is a root of 0

*+

22

+3

=

x

x

x

, then other two roots are AAAAAAAAAAAA.

153.

0

0

0

yzxz

zyxy

zxxyz

B AAAAAAAAAAAAAAA.

8/11/2019 3.3 Exercise

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8/11/2019 3.3 Exercise

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1*1. 6atrices of different order can not be s$btracted.

1*2. 6atrix addition is associative as well as comm$tative.

1*3. 6atrix m$ltiplication is comm$tative.

1*&. s/$are matrix where ever( element is $nit( is called and identit( matrix.

1*5. IfAandBare two s/$are matrices of the same order, then ABBA +=+ .

1**. IfAandBare two matrices of the same order, then ABBA = .

1*+. If matrix OAB = , then OA = or OB = or bothAandBare n$ll matrices.

1*8. #ranspose of a col$mn matrix is a col$mn matrix.

1*. IfAandBare two s/$are matrices of the same order, then BAAB = .

1+0. If each of the three matrices of the same order are s(mmetric, then their s$m is a

s(mmetric matrix.

1+1. IfAandBare an( two matrices of the same order, then ( ) BAAB = .

1+2. If ( ) ABAB = , whereAandBare not s/$are matrices, then n$mber of rows in A

is e/$al to n$mber of col$mns in B and n$mber of col$mns in A is e/$al to

n$mber of rows inB.

1+3. If A, Band Care s/$are matrices of same order, then ACAB = alwa(s implies

that CB = .

1+&. AA is alwa(s a s(mmetric matrix for an( matrixA.

1+5. If

=

2&1

132A and

=

12

5&

32

B , thenABandBAare defined and e/$al.

1+*. IfAis sew s(mmetric matrix, then 2A is a s(mmetric matrix.

1++. ( ) 111

. = BAAB , where A and B are invertible matrices satisf(in'

comm$tative propert( with respect to m$ltiplication.

1+8. If two matricesAandBare of the same order, then ABBA 22 +=+ .

1+. 6atrix s$btraction is associative.

180. )or the non sin'$lar matrix ( ) ( )= 11, AAA .

8/11/2019 3.3 Exercise

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181. CBACAB == for an( three matrices of same order.

182. ( ) ( ) 3113 = AA , whereAis a s/$are matrix and 0A .

183. ( ) 11 1

= Aa

aA , where ais an( real n$mber andAis a s/$are matrix.

18&. 11 AA , whereAis nonsin'$lar matrix.

185. If A and B are matrices of order 3 and 3,5 == BA , then

&05352+3 ==AB .

18*. If the val$e of a third order determinant is 12, then the val$e of the determinant

formed b( replacin' each element b( its cofactor will be 1&&.

18+. 0

&3

32

21

=

+++

+++

+++

cxxx

bxxx

axxx

, then where cba ,, are in ..

188. 2

. AAadj = , whereAis s/$are matrix of order two.

18. #he determinant

BCAC

BBAB

BAAA

cossincossin

cossincossin

cossincossin

++

+

is e/$al to -ero.

10. If the determinant

$nwrcz

%&qby

#'(pax

+++

+++

+++

splits into exactl( ) determinants of

order 3, each element of which contains onl( one term, then the val$e of) is 8.

8/11/2019 3.3 Exercise

46/47

11. et 1*==

zrc

yqb

xpa

, then 321

=

+++

+++

+++

=

rczczr

qbybyq

paxaxp

.

12. #he maxim$m val$e of

cos111

1"sin1!1

111

+

+ is2

1.

13. #he determinant

( ) ( )

yxx

yxx

yyxyx

cossincos

sincossin

2cossincos

++

= is dependent ofxonl(.

1&. #he val$e of

2

&

2

2

2

1

&

1

2

1

111

CCC

CCC

nnn

nnn

++

++is 8.

8/11/2019 3.3 Exercise

47/47

15. If

=

z

y

x

A

11

32

25

, 201023,80 =++= zyxxyz , then

=

8100

0810

0081

.AadjA .

1*. If

=

=

2

1

2

1

2

33

2

1

2

5&

2

1

,

132

21

310

1

y

AxA thenxB 1,yB 1.