* is a row matrix * An mxn matrix is a square matrix if and only if m=n * A= is a diagonal matrix if = 0 i

Transcription

1 CET MATRICES *A matrix is an order rectangular array of numbers * A matrix having m rows and n columns is called mxn matrix of order * is a column matrix * is a row matrix * An mxn matrix is a square matrix if and only if m=n * A= is a diagonal matrix if = 0 i * A= is a scalar matrix = 0 i =k some constant when i = j 1

2 * A is zero matrix all its elements are zeros * If A and B be the two matrices of same order then = for all possible i and j *KA = K = * -A = (-1)A * A + B = B+A * K(A+B) = KA+KB * A+(B+C)= (A+B)+C * A-B = A+(-1)B 2

3 *A= B= then AB = C = * A(BC) = (AB)C * A(B+C) = AB+AC * (A+B)C = AC + BC * A= then A 1 = * (A 1 ) 1 = A * (KA) 1 = KA 1 * (A+B) 1 = A 1 +B 1 * (A-B) 1 = A 1 -B 1 * (AB) 1 = B 1 A 1 * A is a symmetric matrix if A 1 = A * If A is a skew-symmetric matrix if A 1 = -A 3

4 *Any square matrix can be expressed as sum of symmetric and skew-symmetric matrix * Elements operation of the matrix * If A and B be the two square matrix such that AB = BA = I A is inverse of B i.e A= B -1 B is inverse of A i.e B = A -1 *Inverse of the square matrix if it exists is unique 4

5 1) If A+ 2B = and 2A B = then B = 1) 2) 3) 4) None 5

6 2A +4B = 2A-B = Sub. 5B = B= Answer is 3 6

7 2) If A = 1) 2) 3) 4) None 7

8 X = 3B -2A X = X = Answer is 1 8

9 3) If 1) 1 2) 2 3) 0 4) -2 Solution : Answer is 1 x = 1 9

10 4)If A +2B = 1) 2) 3) 4) 10

11 Solution : 3A + 6B = 4A + 6B = -A = Answer is 4 11

12 5) What must be the matrix X if 2X + 2) 3) 4) 12

13 Solution: given X = Answer is 1 13

14 6) If then a + b + x+y =? 1)5 2) 10 3) 20 4) 0 14

15 Solutions: a=4 b= 6 x 3 = 1 y + 2 = 1 x =1 y = -1 hence a + b + x + y = = 10 Answer is 2 15

16 7) then (x, y) =? 1)(1,0) 2) (1,1) 3) (1,2) 4) ( 2,1) Solutions : x+y =1 y + 1 =1 x-y =1 y = 0 2x = 2 X = 1 Answer is 1 16

17 8) If A = diag. then A 3 = 1) diag. 2) diag. 3) diag. 4) diag. Solutins: A = diag. A 2 = diag. A 3 = diag. Answer is 4 17

18 9) If a matrix A = is 1) Scalar matrix 2) Diagonal matrix 3)Unit matrix 4) Square matrix Solution: By definition above example is square matrix Answer is 4 18

19 10) If A = then (A-2I) ( A 3I) = 1)A 2) I 3) 5I 4) 0 Solution: (A-2I) (A-3I) = = Answer is 4 19

20 11) If A = and B = then 1) Only AB defined 2) Only BA defined 3)AB and BA defined 4) AB and BA not defined 20

21 AB = BA = both AB and BA defined answer is 3 21

22 12) A square matrix such that A 2 = A then What is the value of ( I +A ) 2-3A =? 1) I 2) A 3) A 2 4) 2A Solution: ( I +A ) 2-3A = I+A 2 +2AI -3A = I +A +2A -3A= I Answer is 1 22

23 13) If A is the square matrix such that A 2 = I then (A I) 3 + (A +I) 3-7A = 1) A 2) A 3) I-A 4) I+A Solution: (A I) 3 + (A +I) 3-7A = A 3 I -3A (A-I) + A 3 +I +3A ( A+I) -7A = A -3A 2 +3A +A +3A 2 +3A -7A = A Answer is 1 23

24 14) Let A = (A+I) 50-50A = then the value of b + c + d+ e =? 1) 2 2) -2 3) 4 4) None of these 24

25 Solution : A 2 = ( A+I) 2 = A 2 +2A I + I 2 = 2A + I ( A+I) 3 = 3A+ I ( A+I) 50-50A = 50A + I 50A = I b+ c+ d+e = 2 answer is 1 25

26 15) If A is a square matrix such that A 2 = A Then (I+A) 3-7A =? 1) A 2) I-A 3) I 4) 3A Solution: (I+A) 3-7A A 2 = A = I + A 3 + 3A (I+A) -7A A 3 = A = I + A+ 3A +3A -7A AI =A = I Answer is 3 26

27 16) If A = is a square matrix 1) m n 2) m n 3) m=n 4) None Solution : by definition of the square matrix m = n Answer is 3 27

28 17) The number of all possible matrices of order 3X3 with each entry 0 and 1 is 1) 27 2) 18 3) 81 4) 512 Solution : Since matrix of order 3X3 = 9 and two entries 0 and 1 then number of possible order is = 2 9 = 512 Answer is 4 28

29 18) If A = then A+A 1 = I if the value of is 1) 2) 3) 4)π Solution : A+A 1 = I Answer is 1 29

30 19) If A = then A A 1 = 1) I 2) A 3) 2A 4) 0 Solution : AA 1 = = Answer is 1 30

31 20) If A = then A 2 = 1) 2) 3) 4) 31

32 A 2 = AA = Answer is 3 32

33 21) If F(x) = then 4 F(x 3 )-3F(x) 2) 3) 4) None of these 33

34 Solution: 4F(x 3 )-3F(x) = 4 cos 3 x 3cosx = cos3x 4F(x 3 )-3F(x) = 4sin 3 x 3 sinx = - sin3x cos 3 sin3 0 sin3 cos Answer is 1 34

35 22) If X then the value of X and Y 1) X=5, Y= 4 2) X=-4, Y=5 3)X=4, Y =5 4) X =4, Y = -5 Solution: X- 2Y = -6 4X +3Y = 31 by solving above equation X = 4 Y = 5 Answer is 3 35

36 23) If A = then F(A) = A 2-2A +3I find F(A) 1) 2) 3) 4) 36

37 Solution : A 2 = f(a) = A 2-2A +3I = - = Answer is 4 37

38 24) If A = then A 2 = 1) 2) 3) 4) 38

39 A 2 = AA = = = Answer is 4 39

40 25) If A = then A 3 = 1) A 2) 3A 3) 9A 4) 27A Solution: A 2 = AA= A 3 = A 2 A = 3AA= 3A 2 = 3(3A) = 9A Answer is 3 40

41 26) In a square matrix A = we find that = for all i and j then A = 1)symmetric matrix 2) triangular matrix 3) transpose matrix 4) skew-symmetric matrix Solution : by definition of the symmetric matrix A = A 1 = for all i and j Answer is 1 41

42 27) If the matrix A is both symmetric and skewsymmetric then 1)A is a diagonal matrix 2) A is a zero matrix 3) A is square matrix 4) A is a scalar matrix Solution : A 1 =A and A 1 = -A then A = -A Answer is 2 2A= 0 A = 0 42

43 28) If A = such that A 2 = I 1) =0 2) =0 3) =0 4)

44 A 2 = = 1- Answer is 3 44

45 29) then (x, y) = 1) (1,20) 2) (15,1) 3) (1,15) 4) (15,15) Solution : -2x +3y = -27 x+ 5y = 20 by solving above equation x = 15 y =1 (x,y)= (15,1) Answer is 2 45

46 30) If A = then A 4 + A 3 A 2 = 1) 0 2) 1 3) A 4) A 2 Solution : A 2 = AA = = =I A 3 = A 2 A= IA= A A 4 = A 2 A 2 = IXI= I A 4 + A 3 A 2 =I+A-I= A Answer is 3 46

47 31) If A and B is the square matrix of same order n then (A B) 2 = 1) A 2 AB BA +B 2 2) A 2-2AB +B 2 3) A 2 B 2 4) A 2-2BA +B 2 Solution : (A-B) 2 = (A-B )(A-B) = A 2 AB BA + B 2 Since AB BA in general Answer is 1 47

48 32) If then (x, y )= 1) (-1,3) 2) (3,-1) 3) (1,-3) 4)(1,1) Solution: x +y = 2 -x +y = 4 solving above equation x = -1 and y= 3 Hence (x, y)= (-1,3) Answer is 1 48

49 33) If A= and B = then AB = 1) 2) 3) 4) None 49

50 Solution : AB = = Answer is 2 50

51 34) If A and B be the matrix of order n such that A 2 B 2 = (A-B )(A+B) which of the following will be true 1)Either A or B is zero matrix 3)AB =BA 2) A=B 4) Either A or B is an identity element Solution : (A-B )(A+B) = A 2 +AB BA B 2 = A 2 B 2 If AB = BA Answer is 3 51

52 35) If A = then A 2 = 1) A 2) 2A 3) I 4) 0 Solution : A 2 = AA = = = I Answer is 3 52

53 36) If AB = A and BA = B then B 2 +B = 1)2A 2) 0 3) 2I 4) 2B Solution : B 2 = BB = (BA)B = B(AB) =BA= B B 2 +B =B+B = 2B Answer is 4 53

54 37) If A and B be the two square matrices of same order such that AB = B and BA =A then A 2 +B 2 = 1) 1 2) A +B 3) 2AB 4) 2BA Solution : A 2 +B 2 = AA+BB = A(BA) + B(AB) Answer is 2 = (AB)A + (BA)B = BA + AB = A+ B 54

55 38) If A and B be the two symmetric matrices of same order then (AB-BA) = 1)skew-symmetric matrix 2) symmetric matrix 3)zero matrix 4) identity matrix Solution : A 1 = A, B 1 =B, (AB-BA) 1 = (AB) 1 (BA) 1 = B 1 A 1 -A 1 B 1 Answer is 1 = BA AB = -(AB BA) 55

56 39) If the matrix A = is 1) diagonal matrix 2) symmetric matrix 3)skew-symmetric matrix 4) scalar matrix Solution : if A 1 = -A= is a skewsymmetric matrix Answer is 3 56

57 40) If the matrix A = is known as 1) symmetric matrix 2) upper triangular matrix 3)diagonal matrix 4)skew-symmetric matrix Solution: A 1 = -A = is skewsymmetric matrix Answer is 4 57

58 41)If the matrix is a skewsymmetric matrix then a+b+c +d+e+f = 1) 4 2) 0 3) -4 4) 10 Solution : A 1 =-A = a+b+c +d+e+f = = -4 answer is 3 58

59 42) If A = m,n N then A m +A n = 1) 2A 2) A 3) 0 4) (m+n)a Solution: A 2 = AA= = = A A m +A n = A+A = 2A Answer is 1 59

60 43) Matrix theory was introduced by 1) Newton 2) cayleyhamilton 3)Couchy 4) Euclids Solution : answer is 2 60

61 44) If A and B be the symmetric matrix of same order then (AB 1 -BA 1 ) is 1) skew-symmetric matrix 2) symmetric matrix 3)zero matrix Solution : (AB 1 -BA 1 ) 1 = (AB 1 ) 1 (BA 1 ) 1 4) unit matrix = B A 1 A B 1 = -( AB 1 -BA 1 ) Is a skew-symmetric matrix Answer is 1 61

62 45) if O(A) = 2X3, O(B) = 3X2 and O(C) = 3X3 which of the following is not defined 1) BAC 2) AC +A 1 3) C (A+B) 4) C( A+B 1 ) 1 Solution : 1), 2) 4) are defined C(A+B) is not defined Hence answer is 3 62

63 46) A matrix A is satisfy the equation A = 1) 2) 3) 4) none 63

64 Solution : 1),2),3) are not the solution hence 4 is the solution 64

65 47) if A 2 A +I =0 then the inverse of A is 1) A -2 2) A+I 3) I-A 4) A Solution : A 2 A +I =0 operate A -1 on both sides A -1 A 2 A -1 A +A -1 I =0 A I +A -1 = 0 A -1 = I-A Answer is 3 65

66 48) if A and B will be inverse of each other if and only if 1) AB = BA 2) AB= BA=0 3) AB = 0 and BA=I 4) AB = BA = I Solution : given A = B -1 and B = A -1 AA -1 = A -1 A =I Answer is 4 66

67 49) If A,B,C are the inverses of matrix then (ABC) -1 = 1) A -1- B -1 C -1 2) B -1 A -1 C -1 3) C -1 B -1 A -1 4) None Solution : (AB) -1 = B -1 A -1 (ABC) -1 = C -1 B -1 A -1 Answer is 3 67

68 50) If the matrix = A +B Where A is symmetric and B is skew-symmetric then B = 1) 2) 3) 4) 68

69 Solution : B= = = Answer is 4 69

70 51) If for the any real number let = then for real number and then. = 1) 2) 3) 4) none Solution : answer is 2 70