Matrices

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MATRICESINDO-ASIAN PU COLLEGE 1

MATRICES

SYNOPSIS

Matrix :A rectangular array (arrangement) ofnumbers real or complex is called a Matrix. Thehorizontal lines are called rows and the verticallines are called columns. A set of mn numbersarranged in m rows and n columns is calledm x n matrix.

Types of Matrices

Row & Column Matrices :A matrix having onlyone row is called a row matrix, and matrix havingonly one column is called column matrix.

Zero Matrix :A matrix having all its elements aszeros then it is called a zero matrix or null matrix,it is denoted by 'O'.

Square Matrix : If in a matrix, the number ofrows is equal to the number of columns, then it iscalled a square matrix.

Diagonal Matrix : In a square matrix, theelements a

11, a

22, ...... a

nnare called the elements

of the principal diagonal. If in a matrix all theelements above and below the principal diagonal

are zero then it is called a diagonal matrix. Scalar Matrix :

A diagonal matrix in which allthe principal diagonal elements are equal is calledas scalar matrix.

[4],

20

02,

a

a

a

00

00

00

are scalar matrices of

order 1,2 and 3 respectively.

Unit Matrix (Identi ty Matrix)

:A scalar matrix inwhich each diagonal element is unity is calledthe unit matrix (identity matrix)

I1= [1], I

2 =

10

01, I

3=

100

010

001

are the

unit matrices of order 1,2 and 3 respectively. Upper Triangular Matrix :A square matrix

A = [ aij ] is called upper triangular matrix if

aij= o whenever i > j

Eg: A =

400

750

321

Lower Triangular Matrix :A square matrix A =

( aij) is lower triangular Matrix if 0ija , whenever

i j . Eg.

531

021

001

Equality of Matrices :

Two matrices A and B are equal if :i. they are of the same type (order)ii.each element of A is equal to the corresponding element of B.

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Trace :

The sum of the principal (main) diagonal elementsi.e., a

11+ a

22+ ........ + a

nnof a square matrix A is

called the trace of A :Trace A = Tr (A) = a

11+ a

22+ ........ + a

nn.

i) If A and B are two matrices of order n then Tr (A +B) = Trace A + Trace B. Tr (A -B) = Trace A - Trace B.Tr (kA)=k(Tr(A)) Tr (AT) = Tr(A)ii)If A, B,C are square matrices of order n, then Tr (ABC) = Tr (BCA )= Tr (CAB) = Tr (ACB) = Tr (BCA) = Tr (CBA)iii) Trace of skew -symmetric matrix is Zero.

* iv) Trace is also called as spur.Addi tion o f Matri ces :

If A = (aij)

mxn and

B = (bij)

mxnthen A + B = (a

ij+ b

ij)

mxnAddition is

defined between matrices of the same order. Addition of matrices is both commutative andassociative, i.e., A + B = B + A ( Commutativelaw) and ( A+B) +C = A + (B+C) ( associativelaw).

If every element of the matrix A is multiplied by ascalar k then the matrix obtained is written askA. If A = (a

ij)

m x nthen

kA= (kaij)

m x n. If A and B are matrices of the

same type then, k(A+B) = kA + kB. Addi tive Inverse :If A is a m x n matrix then the

zero matrix of the type m x n is called the additiveidentity, then -A is called the

additive inverse of A.Product of Matrices :

If A = [ai j]

m x n where 1 i m, 1 j n

and B = (bjk

)n x p

where 1 j n, pk1then the product A B is an m x p matrix and ABis given by

AB=C=(cik)mxp

where Cik=

n

j

jkijba1

cik = a

i1b

1k+ a

i2b

2k+ ......+ a

inb

n k

Matrix multiplication does not follow commutative

law. Matrix multiplication is associative i.e., (AB)C =A(BC).

Matrix multiplication is distributive over matrixaddition i.e., A(B+C) = AB + AC &(B + C)A = BA + CA.

The cancellation law need not hold in matrixmultiplication, i.e., if A, B, C are three matricesthen AB = AC need not imply that B = C. Forexample let

A =

02

01, B =

11

00and C =

23

00.

Then AB = AC =O. But B C Commute :Two matrices A and B commute if

AB = BA.

Transpose of the Matrix :The matrix obtained by interchanging the rowsand columns of a matrix A is called the transposeof the matrix A & if order of A is mxn then order oftranspose of A is n x m, it is denoted by AT.

(AT) T = A( A + B)T= AT+ BT

(AB)T = BTAT

(KA)T = KAT ( K is a scalar)

Special Type of Matrices :

Idempotent : A square matrix is called

idempotent if A2

= A |A| = 0 or 1 Involutary :A square matrix is called involutaryif A2= I |A| = 1

Orthogonal Matrix:A sqare matrix is A is called

an orthogonal matrix if T TAA A A I or 1 TA A

Eg:cos sin

sin cos

if A is an orthogonal matrix A 1 thus

every orthogonal matrix is non - singular.consequently every orthogonal matrix is invertibleResult: If A,B are n n orthogonal matrices, then

AB and BA are also orthogonal matrices.

if A is an orthogonal matrix, then TA1

A and also orthogonal

Nilpotent :A square matrix is called nilpotentmatrix if there exists a positive integer 'n' suchthat

An= O. If 'm' is the least positive integer suchthat Am = O, them 'm' is called the index of thenilpotent matrix.

Every nilpotent matrix is a singular matrix. Conjugate of Matrix :The conjugate of a matrix

A is the matrix obtained by replacing theelements by their corresponding conjugate

complex numbers. It is denoted by A .

Eg : If A =

2 3 7

0 4 2 5

7 6

i i i

i i

i i

then

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ii

ii

iii

A

67

5240

732

If det A = i then det A = i Symmetric Matrix :A square matrix A is called

a symmetric matrix if AT = A.i) A+AT, AAT, ATA are Symmetric matricesii) If A is symmetric then Anis also symmetric for

all nNSkew - Symmetric Matrix :

A square matrix A is called skew - symmetric ifAT = - A.i) A-ATand AT-A are skew - symmetric matricesii) If A is skew - symmetric then Anis symmetric whenever n is an even +ve

integer Anis skew symmetric whenver n is an odd +ve integer .iii) If A is a skew - symmetric matrix of odd order then det A = 0 and that of even order is a perfect square.

If A is a square matrix then

A =222

AAwhere

AAAA TTT

is symmetric

matrix and 2

TAA is a skew - symmetric

matrix.

Hermitian :A square matrix A is called Hermitian.

If the transpose conjugate of A is itself, i.e.,

TA = AASkew - Hermintian : A square matrix is called

skew - Hermintian if TA = - A.

CONCEPTUAL QUESTIONS

1. A square matrix (aij) in which a

ij= 0 for ji and

aij= k (constant) for i = j is

1) Unit matrix 2) Scalar matrix3) Null matrix 4) Diagonal matrix

2. If A = [aij] is a scalar matrix of order n n,such that a

ij= k for all i=j, then trace of A =

1) nk 2) n+k 3)n

k4) 1

3. If A and B are two matrices such that A hasidentical rows and AB is defined. Then AB has1) no identical rows 2) identical rows3) all of its zeros 4)cannot be determined

4. If AB = O, then1) A = O 2) B = O3) A and B need not be zero matrices

4) A and B are zero matrices5. If A and B are two square matrices of order n and

A and B commute then for any real number K.Then1) A - KI, B - KI Commute2) A - KI, B - KI are equal3) A - KI, B - KI do not commute4) A + KI, B - KI do not commute

6. Anxn

and Bnxn

are diagonal matrices thenAB =.................. matrix1) square 2) diagonal3) scalar 4) rectangular

7. If A =

cfg

fbh

gha

, then A is

1) a nilpotent 2) an involutory3) a symmetric 4) an idempotent

8. If A is a symmetric or skew-symmetric matrixthen A2 is1) symmetric 2) skew-symmetric3) Diagonal 4) scalar

9. Let A be a square matrix. consider1) A + AT 2) AAT 3) ATA 4) AT+A5) A - AT 6) AT- A , Then1) all are symmetric matrices2) (2),(4),(6) are symmetric matrices3) (1),(2),(3),(4) are symmetric matrices & (5),(6) are skew symmetric matrices4) 5,6 are symmetric

10. If A, B are symmetric matrices of the same orderthen AB-BA is1) symmetric matrix 2) skew symmetric matrix3)Diagonal matrix 4) identity matrix

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11. If a matrix A is both symetric and skew-symetricthen A is1) I 2) O 3) A 4) Diagonal matrix

12. If A is skew-symmetric matrix and n is oddpositive integer, then Anis1) a symmetric matrix

2) skew-symmetric matrix3) diagonal matrix4) triangular matrix

13. If A is skew-symmetric matrix and n is evenpositive integer , then An is1) a symmetric matrix2) skew-symmetric matrix3) diagonal matrix4) triangular matrix

14. If A, B are two idempotent matrices andAB = B A = 0 then A+B is1) Scalar matrix 2) Idempotent matrix3) Diagonal matrix 4) Nilptent matrix

15. If3 3ij

A a is a square matrix so that

2 2 ,ija i j then A is a1) unit matrix 2) symmetric marix3) skew symmetric matrix 4) orthogonal matrix

16. If D1and D

2are two 3 x 3 diagonal matrices then

1) D1D

2is a diagonal matrix

2) D1+ D


3) D12+D


4) 1, 2, 3 are correct

EQUALITY OF MATRICES

17. If ,23

70

41

23

azaz

xyx(x+y+z+a) =

1) -1 2) 0 3) 1 4) 8

18. If

3

34

12

52

z

y

r

r, then

1) r = y = z 2) r = -y = z

3) -r = y = z 4) r = y = -z

19. If A=0 2

3 4

, KA=0 3

2 24

a

b

then arrange the

values of k,a,b, in ascending order

1) k, a, b 2) b, a, k 3) a, k, b 4) b, k, a

Trace of Matrix :

20. If Tr (A) = 6 Tr (4A) =1) 3/2 2) 2 3) 12 4) 24

21. If Tr (A) = 2 + i Tr[ (2-i) A] =1) 2 + i 2) 2 - i 3) 3 4) 5

22. If,

017

654

321

A ...)(,

540

030

001

BATrB

1) 40 2) 45 3) 39 4) 5

23. If Tr (A) = 8 , Tr(B) = 6, Tr (A - 2B) =1) -4 2) 4 3) 2 4) 11

24.

6 10 100

7 1 0 ( )

0 9 10

TIf A then Tr A

1) -17 2) 17 3) -1/17 4) 1/17

25. If the traces of the matrices A and B are 20 and 8,then trace of A + B = (EAMCET-1992)1) 28 2) 20 3) - 8 4) 12

26.

017

654

321

A

,

,

540

030

001

B

)(ABTr Tr (A).Tr (B) then =

1) 1 2)0 3)6

54)

20

27

27. i. Trace of the matrix is called sum of the elementsin a principle diagonal of the square matrix.

ii. The trace of the matrix

8 7 2

5 8 2

7 2 8

is 24

Which of the following statement is correct.1. Only i 2. Only ii3. Both i and ii 4. Neither i nor ii

Sum and dif ference of the matrices : -

28.

43

21+ 2x =

95

53, X =

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1)

52

322)

251

231

3)

5232

4)

2

512

31

29. If A-2B =1 2

3 0

and 2A-3B =3 3

1 1

then B =

1)-5 7

5 1

2)-5 7

-5 -1

3)-5 7

5 -1

4)-5 -7

-5 -1

30. If A =

11

41

21

12B then

43

74is

1) 2A + B 2) A - B 3) AB 4) A - 2B

31. If A =

116

51

,34

19

B &

3A + 5B + 2X = 0 then X =

1)

3221

14162)

3221

1416

3)

3221

14164)

3221

1416

32. The additive inverse of

1 4 7

3 2 5

2 3 1

is

1)

1 4 7

3 2 5

2 3 1

2)

1 4 7

3 2 5

2 3 1

3) not possible 4)

1 4 7

3 2 5

2 3 1

33. If

i

iA

0

0then AA2 =

1)

10

012)

10

01

3)

10

014)

10

01

34. If

524

321A and

12

54

32

Bthen

1) AB, BA exist and equal2) AB, BA exist and are not equal3) AB exists and BA does not exist4) AB does not exist and BA exists

35. If A =

654

321

, B =

5

0

1

, then AB =

1) 1501 2) 3004

3)

34

164) 3416

36. IAthenAIf 5,40

322

=

1)

160

184

2)

110

181

3)

115

1314)

115

131

37.4

,01

10AthenAIf

= (EAMCET-1994)

1) I 2) 0 3) A 4) 4I

38. If A =

200

020

002

, then AA4

= ....

1) 16A 2) 32I 3) 4A 4) 8A39. If AB = A and BA = B then

1) A = 2B 2) A2= A and B2= B3) 2A = B 4) cannot be determined

40. If

y

xC

bh

haByxA ,,, ,

then ABC=

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1) bxyhyax 2) 22 2 byhxyax 3) 22 2 byhxyax 4) 22 2 ayhxybx

41. If A2= A, B2= B, AB = BA = O then(A+B)2 =

1) A - B 2) A + B 3) A2

- B2

4) 0

42. If

00

10,

10

01EI , then 3)( bEaI

1) bEaI 2) EbIa 33 3) EabIa

233 4) bEaIa 23 3

43. If A =Cos Sin

Sin Cos

then .A A =

1)A 2)A 3)A A 4)I

44. If A+B=3 4

2 5

, A-B1 2

2 3

then AB=

1)0 40

8 16

2)0 40

4 8

3)0 10

2 4

4)0 40

2 11

45. If A = diagonal (3,3,3) then 4A

1)12A 2)81A 3)684A 4)27A

46. If1 -2

A=4 5

and 2 3 7f t t t then

f(A)+3 6

12 9

= (EAMCET-2008)

1)1 0

0 1

2)0 0

0 0

3)0 1

1 0

4)1 1

0 0

47. If AB=A, BA=B and

I) A2

B=A2

II) ABA=A, BAB=BIII) A2=A, 2B BThen which of the above statements is / are correct1) All the three I, II and III 2) only I and II3) only II and III 4) only I and III

48. Let P and Q be 22 matrices. Consider thestatements.

I) PQ=0 P=0 or Q=0 or bothII) PQ=I

2 P=Q-1III) (P+Q)2=P2+2PQ+Q2

1) I and II are false but III is true2) I and III false and II is true3) All are false 4) All are true

Problems based on Induction :

49. ,

xxxxAIf then NnA

n ...,..........

1)

nnnn

nnnn

xx

xx

22

222)

nnnn

nnnn

xx

xx11

11

22

22

3)

nnnn

nnnn

xx

xx22

22

22

224)

1111

1111

22

22nnnn

nnnn

xx

xx

50. I f 'n' is a +ve integer and i f

A =cos sin

sin cosh

h h

h

then AAn =

1)

hh

hh

cossin

sincos 2)

hh

hh

cossin

sincos

3)

nhnh

nhhn

cossin

sincos 4)

nn

nn

sinhsinh

coshcosh

51. Matrix A is such that A2= 2A - I where I is the

unit matrix . Then for n 2, AAn= (EAMCET-1992)

1) InnA )1( 2) InA 3) 12 ( 1)n A n I 4) 12 1 An

52.4 1then ............,

n

i o o

o i o A n N

o o i

1)

100

010

001

2)

100

010

001

3)

i

i

i

00

00

00

4)

i

i

i

00

00

00

53.2 -1

3 -2

n

=1 0

0 1

if n is

1) odd 2) any natural number

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3) even 4) not possible

54. If

01

10A and

0

0

i

iB then

1) A2= B2= I 2) A2= B2 = -I

3) A

2

= I, B

2

= -I 4) A

2

= - I, B

2

= I

55. If A =0

0

i

i

B=

01

10and C =

0

0

i

i

then AB 1) - BA 2) - C 3) BA 4) AB

56. If P =

4

3

1

, Q = 512 , then PQ =

1)

2048

1536

512

2)

2032

3)

20

3

2

4) 19

57. If A =

0

0

a

a, B =

bb

00, then AB =

1) 0 2) bA 3) aB 4) ab AB

58. If

i

iA

0

0, B =

oi

ioC,

01

10

then A2+ B2+ C2=1) I2 2) - I2 3) - I3 4) I3

59. If A =

01

10 then AA5=

1) I 2) O 3) A 4) A2

60. If A =

dc

baand

10

01I then

A2- (a + d) A - ( bc - ad ) I =1) 0 2) I 3) 2I 4) (a - d )

61.

dc

ba

y

x

0

0=

1)

dyyc

bxax2)

dy

ax

0

0

3)

dybx

cyay4)

0

0

dy

ax

62.

nml

zy

x

0

00

c

b

a

00

00

00

=

1)

ncmbal

bzay

ax

0

00

2)

mbal

azab

ax

0

0

00

3)

nc

mbbz

alabax

00

0 4)

alabax

mbbz

nc

0

00

63. If A =

oab

aoc

bco

and B =

2

2

2

cbcac

bcbab

acaba

then

AB =1) A 2) B 3) I 4) O

64. If A =

2

2

2

cbcac

bcbab

acaba

and a2+ b2+ c2= 1,

then A2=1) A 2) 2A 3) 3A 4) 4A

65. If A =

30

21and B = 13 then BA ==

1)

30

03 2) 03 3) 33 4) 30

66. If A =

431

431

431

then A2 =

1) A 2) - A3) Null matrix 4) 2A

67. If

y

x

1

1

62

41 =

227

144, then (x,y) =

1) ( 1,-2) 2) ( 2,1) 3) (3,2) 4) (2,3)

68.

10

01IIf and

00

10E then 332 EI

1) EI 188 2) EI 364 3) EI 368 4) EI 32

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69. If

10 20 30

20 45 60

30 80 91

=

1 0 0

2 1 0

3 4 1

X 0 0

0 5 0

0 0 1

1 2 3

0 1 0

0 0 1

then X =

1) 5 2) 10 3)5

24)

10

3

70. If A=2 1

3 2

then AA5=

1) I 2)A 3)-A 4) 2A

71. A=1 0

0 2

A3-A2= (EAMCET-2005)

1) 2A 2) 2I 3) A 4) I

72. If A =

43

20, KA =

242

30

b

a, then the

values of k, a, b are respectively (EAMCET-2001)

1) -6, -12, -18 2) -6, 4, 9

3) -6, -4, -9 4) -6, 12, 18

73. If A

cossin

sincos)( then

)(A )(A = (EAMCET-1999)

1) )(A)(A 2) )(A)(A

3) )(A 4) )(A

74. The order of [x y z]

z

y

x

cfg

fbh

gha

is

(EAMCET-1994)1) 3 x 1 2) 1 x 1 3) 1 x 3 4) 3 x 3

75. If A =

01

1xand AA2is identity matrix,then x=

(EAMCET-1993)

1) 1 2) -1 3) 1 4) 0

76. A : A,B are two matrices then AB need not beequal to BAR : Matrix multiplication is associativeThe correct answer is1) Both A and R are true R is correct explanation

to A2) Both A and R are true but R is not correctexplanation to A3) A is true R is false4) A is false R is true

77. A: If A=1 1

1 1

;B=2 2

2 2

then AB=0

R: If AB=0 A or B need not be null matricesThe correct answer is1) Both A and R are true R is correct explanationto A2) Both A and R are true but R is not correctexplanation to A3) A is true R is false 4) A is false R is true

78. I f 'n' is a +ve integer and i f

A =cos sin

sin cos

then AAn =

1)cos sin

sin cos

2)cos sin

sin cos

3)

cos sin

sin cos

n n

n n

4)

cos cos

sin sin

n n

n n

79.

2

1

2

12

1

2

1

Athen NnA

n ...,..........

1) I 2) A 3) 1/2A 4) 2A

80. If

11

43A then AAPwhere isNP

1)2 4

1 2

P P

P P

2)

PP

PP

21

421

3)

PP

PP 4214)

pp

p

21

421

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81. If A =

0 0

0 0

0 0

a

a

a

then AAn =

1) a

n.

A 2) a

n-1

.A 3) a

n+1

.A4) a3n

I

82. If the matrix A =1 1

1 1

then AAn+1=

1) 21 1

1 1

2) 2n1 1

1 1

3) 2n1 1

1 1

4) 2n+11 1

1 1

83. If A=

1 1

1 1

then n N then AAn=

1) 2n-1A 2) 2nA 3) nA 4)2n

84. If1 tan

tan 1

1 tan

tan 1

=a b

b a

1) a =1, b = -1 2) a = sec2, b=0

3) a=0, b = sin2 4)a = sin2, b=cos2

85. If n is a natural number and A =5 8

2 3

then

An=

1)6 2 6

2 1 4

n n

n n

2)4 8

2 2 5

n n

n n

3)1 4 8

2 1 4

n n

n n

4)6 8

2 1 4

n n

n n

86. If1 2

0 1A

then nA

1)1

0 1

n

2)2

0 1

n

3)1

0 2

n

4)1 2

0 1

n

Transpose and properties of transpose of matrix

87.

33

64

r

r=

T

r

r

45

25 then r =

1) 1 2) 2 3) 3 4) -1

88. If A =

cossin

sincos

then A . AAT

1) Null matrix 2) A 3) I2 4) AT

89. xthenXAABA TT ,)(1) BT 2) I + B 3) I + BT 4) BTAT

90. If 3A + 4BT =7 -10 17

0 6 31

and 2B-3AT =

1 18

4 6

5 7

then B =

1)

1 3

1 0

2 4

2)

1 3

1 0

2 4

3)

1 3

1 0

2 4

4)

1 3

1 0

2 4

91. Which of the following is not true, if A and B aretwo matrices each of order n x n, then

1) '')'( ABBA 2) '')'( BABA 3) '')'( BAAB 4) ''')'( ABCABC

92. If

074

701

410

Athen AAT=

1) A 2) - A 3) I 4) A2

93.

032

301

210

AIf then A + AAT=

1)

000

002

020

2)

400

030

001

3)

200

020

002

4)

200

020

202

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94. If 5A =3 4

4 x

and A AAT= ATA=I then x=

1)3 2)-3 3)2 4)-2

95. If 2A+BT=2 3

4 7

, AAT-B =4 5

0 1

then A = ----

1)6 31

1 83

2)2 3

1 8

3)2 31

1 82

4) 0

Problems on Order of Matrices :

96. If the order of A is 4 3, the order of B is

4 5 and the order of C is 7 3, then the order

of (A'B)'C' is

1) 4 5 2) 3 7 3) 4 3 4) 5 7

97. If A and B are two matrices such that A + B and

AB are both defined then

1) A and B are two matrices not necessarily of

same order2) A and B are square matrices of same order

3) A and B are matrices of same type

4) A and B are rectangular matrices of same order

98. If a matrix has 13 elements, then the possible

dimensions (orders) of the matrix are

1) 1 13 or 13 1 2) 1 26 or 26 1

3) 2 13 or 13 2 4) 13 13

99. If A is 3 4 matrix 'B' is a matrix such that A'B

and BA1 are both defined then B is of the type

1) 3 4 2) 3 3 3) 4 4 4) 4 3100. If A=( 1 2 3 4) and AB = (3 4 -1)then the order of

matrix B is

1) 23 2) 33 3) 43 4) 13

SPECIAL TYPES OF MATRICES, SYMMETRIC &

SKEW SYMMETRIC MATRICES

101.

27

61

= P + Q, where P is a symmetric & Qis a skew-symmetric then P =

1)

22

132

131

2)

22

132

131

3)

22

12

11

4)

02

132

130

102.

yxA7

7 is a skew-symmetric matrix,

then (x,y) =1) (1,-1) 2) (7,-7) 3) (0,0) 4)(14,-14)

103. If ATBT= CTthen C =1) AB 2)BA 3)BC 4)ABC

104. If A =

2

2

then A is

1) an idempotent matrix 2) nilpotent matrix3) an orthogonal matrix 4) symmetric

105. If A =

25443

5432

43321

ii

iOi

ii

then A is

1) Hermitian2) Skew-Hermitian3) Symmetric 4) Skew-Symmetric

106. If A =

321

431

422

then A is

1) an idempotent marix 2) nilpotent matrix3) involuntary 4) orghogonal matrix

107. A : AB=A, and BA=BAn

+Bn

=A+BR : AB=A, and BA=B A and B are idempotentThe correct answer is1) Both A and R are true R is correct explanationto A2) Both A and R are true but R is not correctexplanation to A3) A is true R is false 4) A is false R is true

5/21/2018 Matrices

11/12


108. Observe the following lists :

List - I List - II

A) If A is a singular 1) (det A)n-1

matrix then adj A is

B) If A is a square 2) an idempotent Matrix

matrix then detA= C) If A2=A then A is 3) Singular

D) If A is square matrix 4) det AT

of type n then det (adj A) =

5) a nil potent matrix

The correct match for list - I from list - II is

A B C D

1. 2 3 1 5

2. 3 4 2 1

3. 4 3 2 5

4. 1 2 3 4

109. Let A and B be 3 3 matrices such that AAT = -

A. BT = B. Then matrix 3AB BA is a skew-symmertric matrix for :

1) 3 2) 3

3) 3 3or 4) 3 3and

110. A skew - symmetric matrix cannot be of rank

1) 0 2) 1 3) greater than 1 4) 2

111. The maximum number of different possible non-

zero entries in a skew -symmatric matrix of or-

der n is

1) 21

2n n 2) 2

1

2n n

3) 2n 4) 2n n

112. If6 8

,2 10

S P Q

where P is a symmetric

& Q is a skew -symmetric matrix then Q=

1)

05

502)

05

50

3)

08

804)

06

60

113. symmetricaisPQPL ,

121

214

532

matrix, Q is a skew-symmetric matrix thenP =

1)

146

417

672

2)

146

417

672

3)

123

212

7

32

72

4)

246

427

674

114. If A=

2 3 2

3 2 1

4 1 5

x x

is a symmetric matrix

then x=1) 0 2)3 3)6 4)8

115. If A =

1 4

1 0 7

4 7 0

x

such that AAT= -A

then x =1)-1 2)0 3)1 4)4

116. Then matrix A=

2

1

2

12

1

2

1

is

1) unitary 2) orthogonal

3) nilpotent 4) involutary117. List - I List - II

A=

2

2

2

1

1

1

1) Symmetric Matrix

is complex cube root of 1

B =

2 2 4

1 3 4

1 2 3

2) Skew symmetric

C=

a h g

h b f

g f c

3) Nil potent Matrix

D=

O c b

c O a

b a O

4) Singular Matrix

5) Idempotent Matrix

Match of list-I from list - I

5/21/2018 Matrices

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A B C D

1. 4 3 1 2

2. 4 3 2 5

3. 4 5 2 3

4. 4 5 1 2

118. If A = [aij] is a square matrix such that

List - I List - II

A. aij= 1 if i=j 1. symmetricmatrix

= O if i jB. a

ij= O if i j 2. skew symmetric

matrix

C. aij= O if i>j 3. unit matrix

D. aij= i2-j2 i, j 4. diagonal matrix

5. upper triangular

matrix

Correct match of List-I from List-II

A B C D1. 3 4 5 1

2. 3 4 5 23. 3 5 4 2

4. 1 5 3 2

KEY

01) 2 02) 1 03) 2 04) 3 05) 106) 2 07) 3 08) 1 09) 3 10) 2

11) 2 12) 2 13) 1 14) 2 15) 3 16) 4 17) 3 18) 4 19) 4 20) 4

21) 4 22) 1 23) 1 24) 2 25) 126) 4 27) 3 28) 2 29) 2 30) 131) 3 32) 4 33) 2 34) 2 35) 336) 2 37) 1 38) 4 39) 2 40) 241) 2 42) 4 43) 1 44) 3 45) 446) 2 47) 1 48) 2 49) 2 50) 351) 1 52) 3 53) 3 54) 2 55) 356) 1 57) 1 58) 3 59) 3 60) 161) 1 62) 1 63) 4 64) 1 65) 366) 3 67) 4 68) 3 69) 2 70) 271) 1 72) 3 73) 4 74) 2 75) 476) 2 77) 1 78) 3 79) 2 80) 281) 2 82) 3 83) 1 84) 2 85) 386) 4 87) 1 88) 3 89) 3 90) 391) 3 92) 2 93) 1 94) 1 95) 196) 4 97) 2 98) 1 99) 1 100) 3101) 1 102) 3 103) 2 104) 2 105) 1106) 1 107) 1 108) 2 109) 1 110) 2111) 4 112) 2 113) 3 114) 3 115) 2116) 3 117) 4 118) 2

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