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UNIT 2: MATRICES AND DETERMINANTS 1. 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 an identity matrix 2. If in a square matrix A=(aij) we find that aij =aji for all i, j, then A is a)symmetric matrix b)traiangular matrix c)transpose matrix d)skew symmetric

a) c)

a) aij =0 for all i>j b)It is not necessary that aij #0 for all i≤j c)A must be square matrix d)The elements in the principal diagonal may or may not be zero. 4. If O(A)=2x3, O(B)=3x2 and O(C)=3x3, which of the following is not defined? a)C(A+B') b)C(A+B')' c)BAC d)CB+A'

8.

d)None of the above

17.

b)



       [ ] 1 4 3 4 8 0 3 0 7

10. If the matrix

is

a)

1 2 3 2 8 0 3 0 7

0 −2 −1 −2 0 −2 −1 −2 0

c)

a b d

2 −3 c 5 e f

11.

If the matrix

[

]

is symmetric,

x is a)1 b)-1 c)0 d) 2 12. Assuming that the sum and products given below are defined which of the following is not true for matrices? a)AB=AC does not imply B=C(A-1 exist) b)(AB)' =B'A' c)A+B=B+A d)AB=O  A=0or B=0 13. What must be the matrix X if 2X+

    1 2 3 4

=

3 8 7 2

18.

?

∣

∣

2

∣

∣

101 103 105 104 105 106 107 108 109

19. D=

= a)1+a+b+c

∣ ∣ x p p

p q x q q x

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

= (2007)

a)(x+p)(x+q)(x-p-q) b)(x-p)(x-q)(x+p+q) c)(x-p)(x-q)(x-p-q) d)(x+p)(x+q)(x+p+q)

is a skew

2 4x5 x2 1

The only correct

a 1 ab ac 2 ab b 1 bc 2 ac bc c 1

d)None

symmetric matrix, then a+b+c+d+e+f= a)1 b)0 c)-4 d)10



b)1+ab+bc+ca c)1+a2 +b2 +c2 d)abc

The symmetric part of the matrix

A=

0 0 −1 0 −1 0 −1 0 0

statement about the matrix A is a)A-1 does not exist b)A=(-1)I c)A is a zero matrix d)A2=I

If A is a skew symmetric matrix , then (A5 +A8)T is equal to a)A5 +A8 b)A5 -A8 c)A8 -A5 c)AT +BT If A is a symmetric matrix and B is a skew symmetric matrix of the same order , then A2 +B2 is a

1 2 4 6 8 2 2 −2 7



16. Let A=

a)symmetric matrix b)skew symmetric matrix c)unit matrix d)None 9.

] ]

a)0 b)I c)A d)None of these

If A is a square matrix then AAT is a)symmetric b)skew symmetric c)scalar matrix d)unit matrix If a matrix A is symmetric and skew symmetric then A is a a)diagonal matrix b)null matrix

c)Unit matrix 7.

1 −3 2 −1 2 −6 d) 4 −2 2 −1 4 and A+2B= 3 2 5 b)

5 0 3 then B= 1 6 2 8 1 2 8 1 −2 a) b) 1 10 1 −1 10 −1 8 1 2 8 −1 2 c) d −1 10 −1 −1 10 −1 2 −1 15. If A= then A4 +A3 -A2 = 3 −2

matrix A=(aij)?

6.

      [ ] [ ] [ ] [ [ ] [ [ ] 1 3 2 −1 2 6 4 −2

14. If 2A+3B=

3. Which of the following is not correct for a traingular

5.

 

20. If a1, a2 , a3 .........an ....... are in GP, then the value of

∣

the determinant

∣

loga n loga n1 loga n2 loga n3 loga n4 loga n5 loga n6 loga n7 loga n8

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

21. The value of the determinant

∣

∣

1 1 1 mC1 m1C1 m2C1 mC 2 m1C2 m2C 2

is equal to

a)1 b)-1 c)0 d)None of these 22. The value of the determinant Δ=

∣

∣

1 ! 2 ! 3! 2! 3! 4! 3 ! 4 ! 5!

is a)2! b)3! c)4! d)5!

∣

∣

1 −1 4 1 0 1 0 0 5 2 3 0 0 −2 2 −3

23.

=a)41 b)51 c)31 d)26

singular? a)2 b)0

[

3 −1x 3 −1 x3 −1

c)3

2 x2 2

]

B=

d)1

∣

∣

1a 1 1 1 1b 1 1 1 1c 1 1 1   a b c

26. The value of

=0, then the value

33. if

1 d)-a-b-c abc x y yz z x x y z x− y y−z z −x

∣

∣

∣

x−a 0 xc

∣

x−b x−c 0

∣

=

a)

=0 is c)

∣

x x1 x  x−1  x1 x x  x−1 x−2  x1 x  x−1

then f(2012) is a)0 b)1 c)200 d)-200 (C07) 29. If l,m,n are pth , qth , rth term of G.P all positive then

∣ [

logl logm logn

∣

p 1 q 1 r 1

equals

a)-1 b)2 c)1 d)0 (AIEEE02) 30. If A=

1 −1 1 2 1 −3 1 1 1

]

, (CET2007)

c1 c2 c3 a1 a2 a3 b1 b2 b 3

[

and

then

2 1 4 4 2 −3 1 1 2

]

the values of minors of the

] [ ] [ ] [ ] [ ]

34. If A=

a)x=0 b)x=c c)x=b d)x=a 28. If f(x)=

1 2x 3x  x−1

a1 b1 c 1 a 2 b2 c 2 a3 b3 c 3

elements of first row are respectively a)7,11,2 b)1,5,6 c)2,11,7 d)11,7,2

a)0 b)(x+y+z)3 c)2(x+y+z)3 d)2(x+y+z)2 27. If a#b#c, a root of the equation

0 x a xb

and B is the inverse of A,

a)B=0 b)B=A2 c)A= -B d)A=B(2008) 32. If A and B are square matrices of order 3 such that|A|=-1. |B| =3 , then |3AB| is equal to a)-9 b)-81 c)-27 d)81

is

a)-1 b)abc c)

∣ ∣ ∣ ∣

31. If A= is

25. If a, b and c are all different from zero and

of

]

then the value of α is a)0 b)2 c)4 d)5

24. For how many values of x in the closed interval [-4,-1], the matrix

[

4 2 2 −5 0  1 −2 3

10B=

[

0 −2 2 0 2 −3 3 −2 4

then A.adj(A) is equal to

8 0 0 0 8 0 0 0 8

b)

5 1 1 1 5 1 1 1 5

0 0 0 0 0 0 0 0 0

d)

6 0 0 0 6 0 0 0 6

(2007)

35. If A(adjA)=5I where I is identity matrix of order 3, then |adjA| is equal to a)5 b)10 c)125 d)25(2008) 36. If A is a square matrix of order 3 and |A|=8; then | adjA|= a)8 b)82 c)83 d)

1 8

37. If A is a 3x3 matrix and adjoint of A is B. If |B|=64, then |A|= a)64 b)4 c)8 d)3 38. If x+3y+z=6, 3x-2y-8z=7 and 4x+5y-3z=17 , By matrix method x,y,z are a)1,5,0 b)5,0,1 c)5,1,0 d)None 39. The equation x+2y+3z=1; x-y+4z=0; 2x+y+7z=1 have (CET97)a)only one solution b)only two solution c)no solution d)infinitely many solution

40. If A2-4A-5I =0, then A-1 =a)4A-Ib)A-4 c)A-4I/5