Matrices & Determinants MCQs Part IV

1) The value of the determinant
a) 0
b) -1
c) 1
d) 10

2)
a) a/b is one of the cube roots of unity
b) a is one of the cube roots of unity
c) b is one of the cube roots of unity
d) a/b is one of the cube roots of -1

3) If A, B and C are the angles of a triangle and
then the triangle ABC is

a) isosceles
b) equilateral
c) right angled isosceles
d) none of these

4) If a b c are the sides of a
ABC and A, B,C are respectively the angles opposite to them, then

a) sin A – sin B sin C
b) abc
c) 1
d) 0

5) If m is a positive integer and
Then the value of

a) 0
b)
c)
d)

6)
a) 0
b) 1
c) 2
d) 4 pqr

7) If x, y, z are in A. P., then the value of the determinant
a) 1
b) 0
c) 2a
d) a

8)
a) 0
b) 1
c) 100
d) -100

9) If a, b, c, are in A. P., then the value of
a) 3
b) -3
c) 0
d) none of these

10) If p + q + r = 0 = a + b + c, then the value of the determinant
a) 0
b) pa + qb + rc
c) 1
d) none of these

11) If
are respectively the
terms of a GP, then the value of the
determinant

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

12) If A is an invertible matrix, then
is equal to

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

13) The value of the determinant
a) 1
b) -1
c) 0
d) none of these

14) The determinant
is divisible by

a) x
b)
c)
d) all the above

15)
a)
b)
c)
d) none of these

16)

The determinant
is equal to zero, if


i ii iii iv
a, b, c, are in AP a, b, c, are in GP a, b, c are in HP


a) (i) and (ii)
b) (i) and (iii)
c) (iv) only
d) (i) and (iv)

17)
a) 0
b) 1
c)
d) none of these

18) If
are in GP, then the determinant

a) 0
b) 1
c) 2
d) none of these

19) If x, y, a are all distinct and
a) -2
b) -1
c) -3
d) none of these

20)
a) 1
b) 2
c) 3
d) 4

21) If A is square matrix of order n such that its elements are polynomial
in x and its r-rows become identical for
then

a)
b)
c)
d)

22)
then the value of k is

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

23)
a) 0
b) abc
c) -abc
d) none of these

24) The value of the determinant
a)
b)
c)
d) none of these

25) If A, B, C are the angles of a triangle, then the value of
a) cos A cos B cos C
b) sin A sin B sin C
c) 0
d) none of these

26)
a)
b)
c)
d) none of these

27) If x, y, z are in AP, then the value of the det A is, where
a) 0
b) 1
c) 2
d) none of these

28) If ,
then the value of x satisfying the equation

a) a
b) b
c) c
d) 0

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

30)
a) 3
b) 2
c) 4
d) none of these

31) The value of the determinant
is

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

32) If
is a non-singular matrix and
is a square matrix, then Det
is equal to

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

33) The values of
and
for which the system of equations
have no solution are

a)
b)
c)
d)

34) One factor of
a)
b)
c)
d) none of these

35) Let A
a)
b)
c)
d) All the above

36) If a, b,c are non-zero real numbers, then vanishes when
a)
b)
c)
d)

37) The value of the determinant
a)
b)
c)
d)

38) The system of simultaneous equations x + 2y – z = 1 , (k-1)y – 2z = 2 and
(k+2)z=3 have a unique solution if k equals

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

39) The value determinant
a)
b) 3
c) -3
d) none of these

40) If a, b, c are non-zero real numbers such that
a)
b)
c)
d) all the above

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

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

42) The determinant
a) a
b)
c)
d)

43) If
is a cube root of unity then a root of the following polynomial
is

a) 1
b)
c)
d) 0

44) If
a)
b)
c)
d)

45)
a) 0
b) 1
c) -1
d) 2

46)
a)
b)
c)
d)

47) The factors of
are

a)
b)
c)
d)

48) If
is a cube root of unity, then

a) 1
b)
c)
d) 0

49)
a)
b)
c)
d)

50) A and B are two non-zero square matrices such that Then,
a) both A and B are singular
b) either of them is singular
c) neither matrix is singular
d) none of these

51) The roots of the equation
a) 1,2
b) -1,2
c) 1,-2
d) -1,-2

52) From the matrix equation
we can conclude
provided

a) A is singular
b) A is non-singular
c) A is symmetric
d) A is square

53) If k is a scalar and A is
square matrix. Then

a)
b)
c)
d)

54)
a) 1
b) 0
c) -1
d) 67

55)
a) 4
b) x + y + z
c) xyz
d) 0

56) A root of the equation
a) a
b) b
c) 0
d) 1

57) Let a, b, c be positive real numbers. The following system of equations
in x, y and z

a) no solution
b) unique solution
c) infinitely many solutions
d) finitely many solutions

58) If
are the cube roots of unity, then
has the value

a) 0
b)
c)
d) 1

59) In a third order determinant, each element of the first column
consists of sum of two terms, each element of the second column
consists of sum of three terms and each element of the third column
consists of sum of four terms. Then it can be decomposed into
n determinants, where n has the value

a) 1
b) 9
c) 16
d) 24

60) A root of the equation
a) 6
b) 3
c) 0
d) none of these

61)
a) x + y
b) xy
c) x – y
d) 1 + x + y

62)
a) a + b + c = 0
b) -(a + b + c)
c) 0, a + b + c
d) 0,- (a + b + c)

63) If A and B are square matrices of order 3 such that
then

equals
a) -9
b) -81
c) -27
d) 81

64)
a)
b)
c)
d)

65) If A is a singular matrix, then A adj A is
a) identity matrix
b) null matrix
c) scalar matrix
d) none of these

66) If
are the roots of the equation
then value of the

a) p
b) q
c)
d) 0

67) A square matrix can always be expressed as a
a) sum of symmetric matrix and a skew symmetric matrix
b) sum of diagonal matrix and a symmetric matrix
c) skew matrix
d) skew symmetric matrix

68) For a square matrix A, it is given that
, then A is a

a) orthogonal matrix
b) diagonal matrix
c) symmetric matrix
d) none of these

69) Choose the correct answer
a) Every scalar matrix is an identity matrix
b) Every identity matrix is a scalar matrix
c) Every diagonal matrix is an identity matrix
d) A square matrix whose each element is 1 is an identity matrix

70) If A and B are square matrices of the same type then
a) A+ B = B+ A
b) A+ B = A – B
c) A – B = B – A
d) AB= BA

71) The value of is
a) 0
b) a + b+ c
c) 4 abc
d) abc

72)
a) a = 1, b = 1
b)
c)
d)

73) If A,B are square matrices of order 3, then
a)
b)
c)
d)

74) A row matrix has only
a) one element
b) one row with one or more columns
c) one column with one or more rows
d) one row and one column

75) A column matrix has only
a) one row and one column
b) one row with one or more columns
c) one column with one or more rows
d) one element

76) If A and B are two matrices such that A+B and AB are both defined,
then

a) A and B can be any matrices
b) A, B are square matrices not necessarily of same order
c) A,B are square matrices of same order
d) Number of columns of A = number of rows of B

77) If A is a square matrix, then adj
is equal to

a)
b)
c) null matrix
d) unit matrix

78) If A=
is a scalar matrix of order nxn such that =k
for all I, then trace of A is equal to

a) nk
b) n+k
c) n/k
d) none of these

79) is
the identity matrix of order n, then rank of
is

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

80) For the equations: x+2y+3z=1,2x+y+3z=2,5x+5y+9z=4
a) there is only one solution
b) there exists infinitely many solution
c) there is no solution
d) none of these

81) From the matrix equation AB = AC we can conclude B=C provided
a) A is singular
b) A is non – singular
c) A is symmetric
d) A is square

82) If
is equal to

a) unit matrix
b) null matrix
c) A
d) -A

83) If
a) AB = BA = 0
b)
c)
d) none of these

84) If then

a)
b) does not exist
c) is a skew symmetric matrix
d) none of these

85) The system of equations 3x + y – z = 0, 5x +2y – 3z= 2, 15x + 6y – 9z
= 5 has

a) a unique solution
b) two distinct solutions
c) no solution
d) infinitely many solutions

86) Matrix Theory was introduced by
a) Newton
b) Cayley-Hamilton
c) Cauchy
d) Euclid

87) If A is singular matrix, then Adj A is
a) Singular
b) Non-singular
c) Symmetric
d) Not defined

88) If A and B are any
matrics, then det (A +B) = 0 implies

a) det A + det B = 0
b) det A = 0 or det B = 0
c) det A = 0 and det B = 0
d) none of these

89) The parameter on which the values of the determinant
does depend upon

a) a
b) p
c) d
d) x

90) The equation x + 2 y + 3 z = 1,x – y + 4 z = 0, 2x + y + 7 z = 1 have
a) only one solution
b) only two solutions
c) no solution
d) infinitely many solutions

91)
a) 0
b) 1
c) i
d)

92) The number of solutions of
is

a) 0
b) 1
c) 2
d) infinitely many

93) If the system of equations
and
has a non-trivial solution, then the value of
is

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

94) The value of a for which the system equation
, ,
has a non- zero solution is

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

95) The matrix is
known as

a) symmetric matrix
b) diagonal matrix
c) upper triangular matrix
d) skew symmetric matrix

96) For a square matrix A, it is given that ,
then A is a

a) orthogonal matrix
b) diagonal matrix
c) symmetric matrix
d) none of these

97) A square matrix can always be expressed as a
a) sum of symmetric matrix and a skew symmetric matrix
b) sum of diagonal matrix and a symmetric matrix
c) skew matrix
d) skew symmetric matrix

98) If A is a square matrix, then is
equal to

a)
b)
c) null matrix
d) unit matrix

99) If A is a non-zero column matrix of order m x 1 and B is a non-zero row
matrix of order 1 x n, then rank of AB is equal to

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

100) If A, B and C are the angles of a triangle and
then the triangle must be

a) Equilateral
b) Isosceles
c) Any triangle
d) Right angled

Answers
Ans 1) c

= 10 – 9 = 1

Ans Desc 1)

= 10 – 9 = 1

Ans 2) d

Ans Desc 2)

Ans 3) a



Ans Desc 3)


Ans 4) d

From the sine rule, we have


Ans Desc 4)

From the sine rule, we have

Ans 5) a
We have:


Ans Desc 5)
We have:

Ans 6) c

Ans Desc 6)

Ans 7) b
Since x,y,z are in A. P. Therefore, x+z-2y=0.
Ans Desc 7)
Since x,y,z are in A. P. Therefore, x+z-2y=0.

Ans 8) a


Ans Desc 8)

Ans 9) c


Ans Desc 9)

Ans 10) a

Ans Desc 10)

Ans 11) b
Let A be the first term and R be the common ratio of the GP. Then,



Ans Desc 11)
Let A be the first term and R be the common ratio of the GP. Then,


Ans 12) b

Ans Desc 12)

Ans 13) a





Ans Desc 13)




Ans 14) d






Clearly,
divisible by
If
then also it can be easily
seen that
is divisible by

Ans Desc 14)






Clearly,
divisible by
If
then also it can be easily
seen that
is divisible by

Ans 15) d

Now,



Ans Desc 15)

Now,


Ans 16) d




x
is a root of the equation

Ans Desc 16)




x
is a root of the equation

Ans 17) a
We have,



Ans Desc 17)
We have,


Ans 18) a


Similarly



= 0. [Using (I), (ii), (iii)]

Ans Desc 18)


Similarly



= 0. [Using (I), (ii), (iii)]

Ans 19) b




Ans Desc 19)



Ans 20) b

We have,








Ans Desc 20)

We have,







Ans 21) a
Since
will be common from each row which vanish by putting
therefore
will be a factor of

Ans Desc 21)
Since
will be common from each row which vanish by putting
therefore
will be a factor of

Ans 22) c




Ans Desc 22)



Ans 23) b

Ans Desc 23)

Ans 24) d
We have
Ans Desc 24)
We have

Ans 25) c

Ans Desc 25)

Ans 26) a








Ans Desc 26)







Ans 27) a


Ans Desc 27)

Ans 28) d
Clearly,
satisfies the equation.

Ans Desc 28)
Clearly,
satisfies the equation.

Ans 29) d
For ,
the determinant reduces to the determinant of a skew-symmetric matrix of
odd order which is always zero. Hence,
is the solution of the given equation.

Ans Desc 29)
For ,
the determinant reduces to the determinant of a skew-symmetric matrix of
odd order which is always zero. Hence,
is the solution of the given equation.

Ans 30) c





Ans Desc 30)




Ans 31) c
We have

Ans Desc 31)
We have

Ans 32) b

Ans Desc 32)

Ans 33) b
We have,X+y+z=6
X+2y+3z=10
X+2y+z=
This system of equations can be written as




If


Ans Desc 33)
We have,X+y+z=6
X+2y+3z=10
X+2y+z=
This system of equations can be written as




If

Ans 34) a
We have



Ans Desc 34)
We have


Ans 35) d
We have





Ans Desc 35)
We have




Ans 36) a


Ans Desc 36)

Ans 37) c


Ans Desc 37)

Ans 38) b
The system of equations

Ans Desc 38)
The system of equations

Ans 39) b


Ans Desc 39)

Ans 40) d




Ans Desc 40)



Ans 41) c
For the non-trivial solution we must have




Ans Desc 41)
For the non-trivial solution we must have



Ans 42) a




Ans Desc 42)



Ans 43) d







Ans Desc 43)






Ans 44) b

Ans Desc 44)

Ans 45) d
We have,




Ans Desc 45)
We have,



Ans 46) b




Ans Desc 46)



Ans 47) a






Ans Desc 47)





Ans 48) d


Ans Desc 48)

Ans 49) a
We observe that
is obtained from D by applying the following operations

Ans Desc 49)
We observe that
is obtained from D by applying the following operations

Ans 50) b


Ans Desc 50)

Ans 51) b





Ans Desc 51)




Ans 52) b



Ans Desc 52)


Ans 53) d

Since k is common from each row of
and there are n rows in A, therefore


 



Ans Desc 53)

Since k is common from each row of
and there are n rows in A, therefore


 


Ans 54) b

Ans Desc 54)

Ans 55) d
Apply
and take
common from
and from
to make first two columns identical.

Ans Desc 55)
Apply
and take
common from
and from
to make first two columns identical.

Ans 56) c
On expanding the given determinant, we obtain
Ans Desc 56)
On expanding the given determinant, we obtain

Ans 57) b

Then the given system of equations is
The coefficient matrix is
Clearly
So, the given system of equations has a unique solution

Ans Desc 57)

Then the given system of equations is
The coefficient matrix is
Clearly
So, the given system of equations has a unique solution

Ans 58) a



Ans Desc 58)


Ans 59) d

Ans Desc 59)

Ans 60) c




Ans Desc 60)



Ans 61) b

Ans Desc 61)

Ans 62) d

Ans Desc 62)

Ans 63) a


Ans Desc 63)

Ans 64) a

Ans 65) b
We know that



Hence A(adj A) is the null matrix.

Ans Desc 65)
We know that



Hence A(adj A) is the null matrix.

Ans 66) d
Since are the roots of



= 0

Ans Desc 66)
Since are the roots of



= 0

Ans 67) a

Ans 68) a
Matrix is orthogonal.
Ans Desc 68)
Matrix is orthogonal.

Ans 69) b

Ans 70) a

Ans 71) c






Ans Desc 71)





Ans 72) b











Ans Desc 72)










Ans 73) c

Ans 74) b
By Definition of row matrix
Ans Desc 74)
By Definition of row matrix

Ans 75) c
By Definition of column matrix
Ans Desc 75)
By Definition of column matrix

Ans 76) c
Since A+B is defined
A, B are of the same type say
Since AB is defined
n = m.
A,
B must be square matrices of the same type.

Ans Desc 76)
Since A+B is defined
A, B are of the same type say
Since AB is defined
n = m.
A,
B must be square matrices of the same type.

Ans 77) c
We know that

Ans Desc 77)
We know that

Ans 78) a
By definition tr (A) = nk
Ans Desc 78)
By definition tr (A) = nk

Ans 79) b
Since,
Ans Desc 79)
Since,

Ans 80) a
Clearly the determinant of the coefficient matrix is non-zero.
Hence the system of given equations has a unique solution.

Ans Desc 80)
Clearly the determinant of the coefficient matrix is non-zero.
Hence the system of given equations has a unique solution.

Ans 81) b
Since,


Ans Desc 81)
Since,

Ans 82) a

Ans Desc 82)

Ans 83) b



Ans Desc 83)


Ans 84) d








Hence (d) is true.

Ans Desc 84)








Hence (d) is true.

Ans 85) c

and
i.e., 15 x + 6 y – 9z = 6
Also 15 x + 6y – 9z = 5 and

the system has no solution.

Ans Desc 85)

and
i.e., 15 x + 6 y – 9z = 6
Also 15 x + 6y – 9z = 5 and

the system has no solution.

Ans 86) b

Ans 87) a





Since A is singular
Hence Adj A is singular.

Ans Desc 87)





Since A is singular
Hence Adj A is singular.

Ans 88) d
Let


But det A = 1, det B = 1



Ans Desc 88)
Let


But det A = 1, det B = 1


Ans 89) b
The value of the determinant





which is independent of p.

Ans Desc 89)
The value of the determinant





which is independent of p.

Ans 90) d
Here





the given
system is consistent and has infinitely many solutions.

Ans Desc 90)
Here





the given
system is consistent and has infinitely many solutions.

Ans 91) a
We have,








Ans Desc 91)
We have,







Ans 92) b


Ans Desc 92)

Ans 93) c
Since given system of equations has a non-trivival solution

Operate


Dividing by (1 -a) (1 – b) (1 – c), we get


Ans Desc 93)
Since given system of equations has a non-trivival solution

Operate


Dividing by (1 -a) (1 – b) (1 – c), we get

Ans 94) c
The system of linear equation will have non-zero solution if






Ans Desc 94)
The system of linear equation will have non-zero solution if





Ans 95) d
The matrix is skew symmetric.
Ans Desc 95)
The matrix is skew symmetric.

Ans 96) a
Matrix is orthogonal.
Ans Desc 96)
Matrix is orthogonal.

Ans 97) a
A square matrix can be expressed as a sum of a symmetric matrix and a skew
symmetric matrix.

Ans Desc 97)
A square matrix can be expressed as a sum of a symmetric matrix and a skew
symmetric matrix.

Ans 98) c
We know that
Therefore
= 0 (null matrix)

Ans Desc 98)
We know that
Therefore
= 0 (null matrix)

Ans 99) c

Let be
two non-zero column and row matrices respectively.
Therefore AB =


 



Ans Desc 99)

Let be
two non-zero column and row matrices respectively.
Therefore AB =


 


Ans 100) b
Given equation reduces to

[All others will vanish on the LHS]
[R2-R1 , R3-R2 + R1 ]
[By sine
formula]
therefore either a = b or b = c or c = a
is isosceles

Ans Desc 100)
Given equation reduces to

[All others will vanish on the LHS]
[R2-R1 , R3-R2 + R1 ]
[By sine
formula]
therefore either a = b or b = c or c = a
is isosceles

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