Matrices & Determinants MCQs Part II

1) If the system of equations x + ay + az = 0, bx + y + bz = 0, cx + cy +
z = 0 Where a, b and c are non-zero and non-unity, has a non- trivial solution,
then the value of

a) zero
b) 1
c) -1
d)

2) If
then the value of is

a) 1
b) 2
c) 4
d) 3

3) If in
then
is

a) 0
b) -D
c) D
d)

4) Inverse of is
a)
b)
c)
d) none of these

5)
a) 1
b) 0
c) 3
d) a + b+ c

6) A and B are square matrices of order
is equal to

a)
b)
c)
d)

7) If
Then
is equal to

a) 0
b) 1
c) 100
d) -100

8) If ,
then

a) 5A
b) 10A
c) 16 A
d) 32 A

9) If
the value of x which satisfies the equation 
is

a) x = a
b) x = b
c) x = c
d) x=0

10) If is
cube root of unity , then

a)
b)
c)
d)

11) If the system of equation x + 2 y – 3 z = 2, ( K+3) z = 3, (2K+1)y+z=2 is inconsistent , then K is
a) -2
b)
c) 1
d) 2

12) If the equations x- Ky – z = 0, Kx- y – z = 0, x+ y – z = 0 has a non zero solution, then the possible values of K are
a) -1, 2
b) 1, 2
c) 0, 1
d) -1, 1

13) The parameter on which the values of the determinantdoes not depend upon
a) a
b) p
c) d
d) x

14) If A, B, C are the angles of a triangle ABC, then the determinant
is equal to

a) 1
b) -1
c) sin A + sin B + sin C
d) none of these

15) If a, b, c are different, then the value of x satisfying
a) a
b) b
c) c
d) 0

16) The value of the determinant
a) k (a + b)(b + c) (c + a)
b)
c) k(a – b)(b – c)(c – a)
d) k(a + b – c)(b + c – a)(c + a – b)

17) If are
any four real numbers, then the determinant

a) 1
b) 0
c) 2
d) 3

18) If B is a non-singular matrix and A is a square matrix, then
is equal to

a) det (B)
b) det (A)
c)
d)

19) If
and nN,
then
is equal to

a)
b)
c) n A
d) none of these

20) If A =is
a scalar matrix of order n x n such that
= k for all I, then is
equal to

a) n k
b) n + k
c)
d)

21) If A is an orthogonal matrix, then is
equal to

a) A
b)
c)
d) None of these

22) If A is an orthogonal matrix, then
a)
b)
c)
d) none of these

23) If =then
t is equal to

a) 33
b) 21
c) 31
d) 23

24) If for AX = B, and
then
X is equal to

a)
b)
c)
d)

25)
a)
b)
c)
d) none of these

26) The value of determinant is
equal to

a)
b)
c)
d)

27) The value of determinant
is
equal to

a)
b)
c)
d)

28) ,
then is
equal to

a)
b)
c)
d) 9 tan x sec x

29) If,
then

a)
b)
c)
d)

30) If then is
equal to

a)
b)
c)
d)

31) The matrix A satisfying the equationA
=is

a)
b)
c)
d) None of these

32) If A is an orthogonal matrix, then equals
a) A
b)
c)
d) None of these

33)
a) D
b)
c)
d) None of these

34) If A and B are two invertible matrices, then the inverse of AB is equal to
a) AB
b) BA
c)
d)

35) If A,B,C are invertible matrices, then
is equal to

a)
b)
c)
d)

36) If
And then
n equals

a) 4
b) 6
c) 8
d) none of these

37) If A=diag (
a)
b)
c) A
d) None of these

38) If A = [aij]
is a skew-symmetric matrix of order n, then is

a) 0 for some i
b)
c) 1 for some i
d)

39) If A and B are matrices of the same order,then
is possible, if

a) AB=I
b) BA = I
c) AB=BA
d) None of these

40)
a) 17
b) 25
c) 3
d) 12

41) If A is a skew-symmetric matrix, then trace of A is
a) 1
b) -1
c) 0
d) None of these

42) If A,B are square matrices of the same order,then adj(AB) is equal to
a) (adj A) (adj B)
b) (adj B) (adj A)
c) adj A+adj B
d) adj A-adj B

43) If A is a square matrix of order and
k is a scalar, then adj(KA) is equal to

a) k adj A
b)
c)
d)

44) If A is a square matrix or order then
adj(adjA) is equal to

a)
b)
c)
d)

45) and

a) sinx cosx
b) 1
c) 2
d) 3

46) If
a)
b)
c) n A
d) none of these

47)
a)
b)
c)
d)

48) If
then the value of k is

a) 3
b) 5
c) 7
d) -7

49) If A,B are two square matrics such that AB= A and BA= B, then
a) A,B are idempotent
b) Only A is idempotent
c) Only B is idempotent
d) None of these

50) The inverse of a symmetric matrix is
a) symmetric
b) skew-symmetric
c) diagonal matrix
d) none of these

51) If
is a scalar matrix of order such
that

a) nk
b) n+k
c)
d)

52) If
is a square matrix of order and
k is scalar, then

a)
b)
c)
d) none of these

53) Let F (a)
, Where
Then
equals

a)
b)
c)
d) none of these

54) If F(x)
then

a)
b)
c)
d)

55)
a) a=1, b=1
b)
c)
d) none of these

56) If matrix A is such that
then
is equal to

a)
b)
c)
d) none of these

57) A and B be
matrices. Then AB=0 implies

a) A=0 and B=0
b)
c)
d) A=0 or B=0

58) Which of the following is incorrect
a)
b)
c)
d)

59) If A is an invertible matrix, then which
of following is correct

a)
b)
c)
d)

60) If for a matrix
where I is the identity matrix, then A equals

a)
b)
c)
d)

61) Let A be a skew-symmetric matrix of odd order, then
a) 0
b) 1
c) -1
d) None of these

62) Let A be a non-singular square matrix. Then
a)
b)
c)
d) none of these

63) Let
be a square matrix,and let
be cofactor of

a)
b)
c)
d) none of these

64)
be a matrix such that

a) rank (A) >1
b) rank (A)=1
c) rank (A)=m
d) rank (A)=n

65) The rank of the matrix
a) 1
b) 2
c) 3
d) 4

66) If
is a matrix of rank r, then

a) r=min (m,n)
b) r < min (m,n)
c)
d) none of these

67) is
a matrix of rank r and B is a square submatrix of order r+1, then

a) B is invertible
b) B is not invertible
c) B may or may not be invertible
d) None of these

68) Let A be a skew-symmetric matrix of even order, then
a) is a square
b) is not a square
c) is always zero
d) none of these

69) If A is a non-singular square matrix of order n, then the
rank of A is

a) equal to n
b) less than n
c) greater than n
d) none of these

70) If A is a non-zero column matrix of order and
B is non-zero row matrix of order
then rank of AB equals

a) m
b) n
c) 1
d) none of these

71) If A and B are two matrices such that AB=B and BA=A, then

a) 2AB
b) 2BA
c) A+B
d) AB

72) If
a)
b)
c)
d)

73) If
a)
b)
c)
d) none of these

74) If
a) 3
b) 5
c) 2
d) 4

75) The system of linear equations x+y+z=2, 2x+y-z=3, 3x+2y+kz=4
has a unique solution if

a)
b) -1c) -2d) k=0

76) If A and B are square matrices of order 3 such that
a) -9
b) -81
c) -27
d) 81

77) If A is
and B=
then AB is said to be a square matrix of order

a)
b)
c)
d)

78) In an upper triangular matrix
the element
for

a)
b)
c)
d)

79) If A and B are arbitrary square matrices of same order, then
a)
b)
c)
d)

80) Let A be a square matrix. Then which of the following is not a symmetric
matric

a)
b)
c)
d)

81) If A is of order
and B is of order ,
then AB is of order

a)
b)
c)
d)

82) Let
Where
then
is equal to

a) i
b)
c)
d) -i

83) If
then

a)
b)
c)
d) none of these

84) If
then x equals

a) 1, 1, 0
b) 0, -1, 1
c) 1, -1, 3
d) 0, 0, 3

85) If ,
one value of x which satisfies the equation.

a) x = a
b) x = b
c) x = c
d) x = 0

86) Which one of the following determinant has its value as zero?
a)
b)
c)
d)

87) The value of
a) a + b + c – 3 abc
b) 3 (a +b) (b + c ) (c + a)
c) (a – b) (b – c) (c – a )
d) (a – b ) (b – c ) (c – a ) (a +b + c )

88) The value of satisfying

a)
b)
c)
d)

89) The repeated factor of the determinant
a) (x – y)
b) (y – z)
c) (z – x)
d) none of these

90) The value of
a)
b)
c)
d)

91) If p, q, r are negative distinct real numbers, then the determinant
a) < 0
b)
c) 0
d) > 0

92) The value of
a) 20
b) -2
c) 0
d) 5

93) One of the roots of
a) 6
b) 0
c) 3
d) -3

94) If
is an imaginary cube root of unity, then

a) 1
b) -1
c) 0
d) none of these

95) The value of det
a)
b) 0
c)
d) none of these

96) The value of the determinant
a) 2 (a+b+c)
b)
c) ab+bc+ca
d) 2 abc (ab+bc+ca)

97) If is
the transpose of a square matrix A, then

a)
b)
c)
d)

98) If a square matrix A has two identical rows or columns, then det A
is

a) 0
b) 1
c) -1
d) none of these

99) Let
a)
b)
c)
d) none of these

100) If A is square matrix such that
then det (A) equals

a) 0 or 1
b) -2 or 2
c) -3 or 3
d) none of these

Answers
Ans 1) c
For the non-trivial solution, we must have



Divide through out by (a-1) (b – 1) (c -1)




Ans Desc 1)
For the non-trivial solution, we must have



Divide through out by (a-1) (b – 1) (c -1)



Ans 2) c



Ans Desc 2)


Ans 3) a

Ans Desc 3)

Ans 4) c





Ans Desc 4)




Ans 5) b



Ans Desc 5)


Ans 6) d


Ans Desc 6)

Ans 7) a





Ans Desc 7)




Ans 8) c






Ans Desc 8)





Ans 9) d




if (clearly)

Ans Desc 9)




if (clearly)

Ans 10) d
Operate









Ans Desc 10)
Operate








Ans 11) b
Since the system of equations



is inconsistent.


Ans Desc 11)
Since the system of equations



is inconsistent.

Ans 12) d
The system of equations



has a non-zero solution if



Ans Desc 12)
The system of equations



has a non-zero solution if


Ans 13) b

The value of the determinant





which is independent of p.


Ans Desc 13)

The value of the determinant





which is independent of p.

Ans 14) d
Its value is 0
Ans Desc 14)
Its value is 0

Ans 15) d
Since for x = 0, the determinant reduces to the determinant of a skew symmetric
matrix of odd order which is always zero. Hence x =0 is the solution of
the given equation.

Ans Desc 15)
Since for x = 0, the determinant reduces to the determinant of a skew symmetric
matrix of odd order which is always zero. Hence x =0 is the solution of
the given equation.

Ans 16) c




= k (a – b) ( b – c ) ( c – a )

Ans Desc 16)




= k (a – b) ( b – c ) ( c – a )

Ans 17) b
The given determinant =

Ans Desc 17)
The given determinant =

Ans 18) b
=

=
=
=det (I) . det A
=1. det A = det A

Ans Desc 18)
=

=
=
=det (I) . det A
=1. det A = det A

Ans 19) b



and so on therefore

Ans Desc 19)



and so on therefore

Ans 20) d


Ans Desc 20)

Ans 21) b
Since A is orthogonal matrix
Therefore
Therefore

Ans Desc 21)
Since A is orthogonal matrix
Therefore
Therefore

Ans 22) b

Ans Desc 22)

Ans 23) b
Putting,
we shall get

=1(0-9)+3(4+6)
=-9+30=21

Ans Desc 23)
Putting,
we shall get

=1(0-9)+3(4+6)
=-9+30=21

Ans 24) a
If AX = B, then


Ans Desc 24)
If AX = B, then

Ans 25) a







=0+0+1=1
Also


is divisible
by x

Ans Desc 25)







=0+0+1=1
Also


is divisible
by x

Ans 26) b


(By taking
common from R2)

(By taking
common from C2)

Operate R3+R2





 



Ans Desc 26)


(By taking
common from R2)

(By taking
common from C2)

Operate R3+R2





 


Ans 27) d


Operate R1-R2

Operate C3+C1




 



Ans Desc 27)


Operate R1-R2

Operate C3+C1




 


Ans 28) b

Operate,
we get



=
=





Ans Desc 28)

Operate,
we get



=
=




Ans 29) c


Multiply C1 by x, C2 by y, C3 by z ,we get

Operate C1+C2+C3, we get



Ans Desc 29)


Multiply C1 by x, C2 by y, C3 by z ,we get

Operate C1+C2+C3, we get


Ans 30) c
We have,

Ans Desc 30)
We have,

Ans 31) c
We have,


Ans Desc 31)
We have,

Ans 32) b
By definition A is orthogonal

Ans Desc 32)
By definition A is orthogonal

Ans 33) b

Let D=

Now, cofactor of
Cofactor of
And, cofactor of


= Diag




Ans Desc 33)

Let D=

Now, cofactor of
Cofactor of
And, cofactor of


= Diag



Ans 34) d




Ans Desc 34)



Ans 35) d
We Have

Ans Desc 35)
We Have

Ans 36) d
Since

[Since K is variable ]


[Since C2 – C1 , C3 – C1 ]


But

Ans Desc 36)
Since

[Since K is variable ]


[Since C2 – C1 , C3 – C1 ]


But

Ans 37) b
We have,








Proceeding in this manner, we have

Ans Desc 37)
We have,








Proceeding in this manner, we have

Ans 38) b
Since
is a skew-symmetric matrix. Therefore


Ans Desc 38)
Since
is a skew-symmetric matrix. Therefore

Ans 39) c
We have





Ans Desc 39)
We have




Ans 40) a
We know that, if
is a square matrix of order ,
then tr(A)=.

Ans Desc 40)
We know that, if
is a square matrix of order ,
then tr(A)=.

Ans 41) c
Since diagonal element of a skew-symmetric matrix are all zero

Ans Desc 41)
Since diagonal element of a skew-symmetric matrix are all zero

Ans 42) b
We Know that

We have(AB) (adjB. Adj A)=A(B adj B)adj A
=




From (i) and (ii)
(AB) (adj AB) = (AB) (adjB. adj A)



Ans Desc 42)
We Know that

We have(AB) (adjB. Adj A)=A(B adj B)adj A
=




From (i) and (ii)
(AB) (adj AB) = (AB) (adjB. adj A)


Ans 43) c
We have







Ans Desc 43)
We have






Ans 44) c
For any square matrix X, we have

Taking X=adj A,we get




Ans Desc 44)
For any square matrix X, we have

Taking X=adj A,we get



Ans 45) b

We know that



It is given that A (adjA) =kI. Therefore, k=1.


Ans Desc 45)

We know that



It is given that A (adjA) =kI. Therefore, k=1.

Ans 46) b

Ans Desc 46)

Ans 47) b
We have

Ans Desc 47)
We have

Ans 48) b
We know that
Ans Desc 48)
We know that

Ans 49) a
We have, AB =A and BA= B








are indempotent
matrices.

Ans Desc 49)
We have, AB =A and BA= B








are indempotent
matrices.

Ans 50) a
Let A be a symmetric matrix. Then




is a
symmetric matrix

Ans Desc 50)
Let A be a symmetric matrix. Then




is a
symmetric matrix

Ans 51) d
We have,
Ans Desc 51)
We have,

Ans 52) a
We have
therefore

Ans Desc 52)
We have
therefore

Ans 53) a
We have


Ans Desc 53)
We have

Ans 54) c
We have,


Ans Desc 54)
We have,

Ans 55) b
We have
Or,
Or,

or,

Ans Desc 55)
We have
Or,
Or,

or,

Ans 56) a
We have



Ans Desc 56)
We have


Ans 57) c
We have
Ans Desc 57)
We have

Ans 58) a
We have,
So,
(a) is not true

Ans Desc 58)
We have,
So,
(a) is not true

Ans 59) d
Since A is invertible, therefore
Thus, (d) is correct

Ans Desc 59)
Since A is invertible, therefore
Thus, (d) is correct

Ans 60) b
Obviously,
satisfies the equation

Ans Desc 60)
Obviously,
satisfies the equation

Ans 61) a
Let A be a skew-symmetric matrix of odd order (2 n+1) say.Since
A is skew-symmetric,therefore



Ans Desc 61)
Let A be a skew-symmetric matrix of odd order (2 n+1) say.Since
A is skew-symmetric,therefore


Ans 62) b
Since A is non-singular, therefore
exists and



)

Ans Desc 62)
Since A is non-singular, therefore
exists and



)

Ans 63) b
Since A is non-singular matrix, therefore




Ans Desc 63)
Since A is non-singular matrix, therefore



Ans 64) b
Let A denote the matrix every element of which is unity.Then
all the 2-rowed minors of A obviously vanish. But A is a non-null
matrix. Hence,
rank of A is 1.

Ans Desc 64)
Let A denote the matrix every element of which is unity.Then
all the 2-rowed minors of A obviously vanish. But A is a non-null
matrix. Hence,
rank of A is 1.

Ans 65) c



We observe that the leading minor of the third order of this
matrix i.e.

Ans Desc 65)



We observe that the leading minor of the third order of this
matrix i.e.

Ans 66) c
It is a direct consequence of the definition of rank
Ans Desc 66)
It is a direct consequence of the definition of rank

Ans 67) b
By def. Of rank,
and so it is not invertible.

Ans Desc 67)
By def. Of rank,
and so it is not invertible.

Ans 68) a
Letbe
a skew symmetric matrix.Then

Ans Desc 68)
Letbe
a skew symmetric matrix.Then

Ans 69) a
Since A is non-singular matrix,therefore

Ans Desc 69)
Since A is non-singular matrix,therefore

Ans 70) c

be two non-zero column and row matrices respectively. We have

Since A and B are non-zero matrices, therefore the matrix AB
will also be a
non-zero matrix. The matrix AB will have at least one non-zero
element
Obtained by multiplying corresponding non-zero elements of A
and B.
All the two-rowed minors of A obviously vanish. But A is a non-zero
Matrix.Hence rank (A) =1

Ans Desc 70)

be two non-zero column and row matrices respectively. We have

Since A and B are non-zero matrices, therefore the matrix AB
will also be a
non-zero matrix. The matrix AB will have at least one non-zero
element
Obtained by multiplying corresponding non-zero elements of A
and B.
All the two-rowed minors of A obviously vanish. But A is a non-zero
Matrix.Hence rank (A) =1

Ans 71) c
We have AB=B and BA=A therefore,


Ans Desc 71)
We have AB=B and BA=A therefore,

Ans 72) a
We have
So

Ans Desc 72)
We have
So

Ans 73) d

matrices in (a),(b),(c) do not tally with

Ans Desc 73)

matrices in (a),(b),(c) do not tally with

Ans 74) b
Since the given matrix is symmetric, therefore

Ans Desc 74)
Since the given matrix is symmetric, therefore

Ans 75) a
The given system of equation has unique solution if

Ans Desc 75)
The given system of equation has unique solution if

Ans 76) a
We have


Ans Desc 76)
We have

Ans 77) b
AB of type
Ans Desc 77)
AB of type

Ans 78) b
By Definition.
Ans Desc 78)
By Definition.

Ans 79) d

Ans Desc 79)

Ans 80) d

is not symmetric

Ans Desc 80)

is not symmetric

Ans 81) b
Order of
Ans Desc 81)
Order of

Ans 82) a

Operate





Ans Desc 82)

Operate




Ans 83) b
Operate
we get

Ans Desc 83)
Operate
we get

Ans 84) d
Operate
and taking out 3-x as common from ,
we get

operate
we get


Ans Desc 84)
Operate
and taking out 3-x as common from ,
we get

operate
we get

Ans 85) d
For x = 0 the given determinant is a skew-symmetric determinant of odd
order
its value = 0

Ans Desc 85)
For x = 0 the given determinant is a skew-symmetric determinant of odd
order
its value = 0

Ans 86) d
Value = 0.
All elts of second row = 0]

Ans Desc 86)
Value = 0.
All elts of second row = 0]

Ans 87) d







Ans Desc 87)






Ans 88) d












Ans Desc 88)











Ans 89) c
Operate
we get the given value


Operate



Ans Desc 89)
Operate
we get the given value


Operate


Ans 90) b
The given value


Ans Desc 90)
The given value

Ans 91) d
Operate


Operate






Ans Desc 91)
Operate


Operate





Ans 92) b
Operate
Ans Desc 92)
Operate

Ans 93) b
Operate
we shall get x = 0 as one root

Ans Desc 93)
Operate
we shall get x = 0 as one root

Ans 94) c
Operate ,
we get the


Ans Desc 94)
Operate ,
we get the

Ans 95) c
Value of the det. =
Ans Desc 95)
Value of the det. =

Ans 96) b



Operate



Ans Desc 96)



Operate


Ans 97) b

Ans Desc 97)

Ans 98) a
Det. A = 0
Ans Desc 98)
Det. A = 0

Ans 99) a
Adj
Ans Desc 99)
Adj

Ans 100) a
Since


Ans Desc 100)
Since

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