Matrices and Determinants

Matrices and determinants

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[UNIT 01. MATRICES AND DETERMINANTS] [QUESTIONS FROM CBSE BOARD PAPERS [1992-]

3 1 01. If A= -1 2 , show that A- 5A + 7I = 0. Hence find A-1.[CBSE 06] 02. Prove that 1+a 1 1 1 1+b 1 = abc( 1+ 1/a + 1/b+ 1/c)

1 1 1+c 03. Solve the following system of equations by Cramers rule: 2x+3y = 10,x+6y= 4. 04. Prove that a a+b a+b+c 2a 3a+2b 4a+3b+c = a. 3a 6a+3b 10a+6b+3c 05. If A = cos sin ,then verify that A . A = I. -sin cos 06. If A = 7 0 and I = 1 0 , then find k so that A = 8A + kI. -1 7 0 1 07. Without expanding the determinant show that a+b+c is a factor of the determinant a b c b c a c a b . 08. Solve the following system of equations by Cramers rule: x-2y=4, -3x+5y+7 = 0. 09. If A= 0 -1 5 and B = 4 3 6 3 -4 1 0 show that (AB) = BA.

7 -6 10. Without expanding the determinant, prove that x+y x x 5x +4y 4x 2x = x

10x+8y 8x 3x 11. If A= 1 -5 and B = 3 1 -2 , verify that (AB) = BA. 7 12.Find a 2x2 matrix, B such that B 1 -2 = 6 0 1 4 0 6


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13. Construct a 2x2 matrix a = [aij] , whose elements are given by aij= (i+2j) 2 14. Using determinants, find the area of the triangle whose vertices are (-2,4), (2,-6) and (5,4 ). Are the given points collinear? 15. Define a symmetric matrix. Prove that for A= 1 2 , A + A is a symmetric matrix where A is transpose of A. 3 4 16. Using determinants, find the area of the triangle with vertices (-3,5) (3,-6)&(7,2). 17. Express the matrix A = 3 -4 as the sum of the symmetric and skew- symmetric matrix. 1 -1 18. Compute the adjoint of the matrix A = 1 2 and verify that A.(adj A)= |A| I. 3 -5 19. For matrix -3 6 0 A = 4 -5 8 ,find 1(A- A), where A is the transpose of the matrix A. 0 -7 -2 2 20. If A= 1 3 2 & B = 2 -1 -1 ,find a matrix C such that A + B+C is a 2 0 2 0 1 -1 zero matrix. 21. Construct a 2X3 matrix whose elements in the ith row and jth column are given by i. aij = 3i- j , ii. 2i + 3j iii. (i-2j)

2 2 2 1 a b+c

22. Without expanding the determinant, prove that 1 b c+a = 0. 1 c a+b 23. If f(x) x-4x+1, find f(A) when A = 2 3 .

1 2 1/a a bc 24. Without expanding the determinant, prove that 1/b b ca = 0. 1/c c ab 25. Find a matrix X such that 2A+ B+ X = 0, where A= -1 2 and B = 3 -2 . 3 4 1 5

26. If A = i 0 and B = 0 i ,show that AB BA. 0 i i 0 27. If A = 2 3 ,prove that A AT is a skew symmetric where AT denotes the 4 5 transpose of A.


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28. If A = 4 1 , prove that A + AT is a skew symmetric where AT denotes the 5 8 transpose of A. -1 29. If A = 2 and B = -2 -1 - 4 , verify that (AB) = BA. 3 30.From the following equation, find the values of x and y: 2 x 5 + 3 4 = 7 14 7 y-3 1 2 15 14 0 ab ac 31.Using properties of determinants, evaluate ab 0 bc . ac cb 0 32. Using properties of determinants show that i. a+x y z ii. x y z x a+y z = a(a+x+y+z) x y z = (x-y)(y-z) x y a+z y+z z+x x+y (z-x)(x+y+z) 33. Find X such that X. -5 -7 = -16 -6 2 3 7 2 34.Solve by crammers rule: 5x-7y+z = 11, 6x-8y-z = 15, 3x+2y 6z =7. 35.Show that A = 5 3 satisfies the equation x-3x-7 = 0. Thus, find A-1.

-1 -2 36. Using properties of determinants show that: i. 1 a a- bc ii. x+4 x x 1 b b-ca = 0. x x+4 x = 16(3x+4) 1 c c-ab x x x+4 iii. 1 x x3 iv. a+b+c -c -b 1 y y3 =(x-y)(y-z)(z-x)(x+y+z) -c b+c+a -a =2(a+b)(b+c)(c+a)

1 z z3 -b -a c+a+b

37. If A = 3 2 verify A- 4A-I=0 where I= 1 0 &0= 0 0 hence find A-1. 2 1 0 1 0 0 38. Using cramers rule solve the following system of equations: 3x-2y = 5, x-3y+3 =0.


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39. Using properties of determinants show that: ( i) a+b+2c a b ii) b+c c+a a+b a b c c b+c+2a b = 2(a + b + c) c+a a+b b+c = 2 b c a c a c+a+2b [D 2006] a+b b+c c+a c a b iii. a-b-c 2a 2a iv. bc bc b+c 2b b-c-a 2b = (a +b +c) ca ca c+a = 0. 2c 2c c-a-b [CBSE D06] ab ab a+b v. x+y x x vi. 1 x+y x+y vii. 43 3 6 5x+4y 4x 2x = x. 1 y+z y+ z =(x-y)(y-z)(z-x) 35 21 4 = 0 10x+8y 8x 3x 1 z +x z+ x 17 9 2 40. Solve the following system of equations by matrix method: i. 3x + 4y + 7z = 14, 2x-y+ 3z = 4, x+ 2y 3z = 0 ii. 2x-z = 3, 5x+y = 7, y+3z = -1 iii. x+2y-3z = 6,3x+2y-2z = 3,2x- y+ z=2. iv.x +y +z =1,x-2y+3z=2, x-3y+5z=3 v. x-y+z=3, 2x+y-z=2, -x-2y+2z= -1. vi. x +y +z = 6,x+2y+3z=14,x+4y+7z=30 vii. x+2y-3z = -4, 2x+3y+2z = 2, 3x-3y-4z = 11 viii. 5x+3y+z=16, 2x+y+3z=19, x+2y+4z=25 ix. 2x+6y =2,3x-z=-8, 2x-y +z+3= 0. x. 2 + 3 +10 = 4 , 4 - 6 +5 = 1 , 6 + 9 - 20 = 2 x y z x y z x y z

xi) 6x-12y+25z = 4, 4x+15y-20z= 3, 2x+18y+15z = 10

xii. x+2y+z=7, x+3z=11, 2x-3y=1 xiii. x +y +z= 3,x-2y+3z=2,2x-y+z=2[D06] xiv. x+2y+z=1; 2x-y+z=5, 3x+y-z=0 xv.2x+y-3z=13; x+y-z=6; 2x-y+4z=-12 xvi. 2x-y+2z=3; 2x+y+z= -1; x-3y+2z = 6.xvii. 2x+y-z=6, x+3y-2z=7, x-y-z = 2. 41.If A = 1 -2 0 2 1 3 , find A-1. Using A-1, solve the following system of equations: 0 -2 1 x-2y= 10, 2x+y+3z=8, -2y+z= 7.


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42. If A = 8 -4 1 10 0 6 , find A-1. Using A-1, solve the following system of equations: 8 1 6 8x-4y+z=5, 10x+6z=4, 8x+y+6z=5/2

1 2 2 43. If A = 2 1 2 ,find A-1 and hence prove that A- 4A 5I = 0. 2 2 1 44. Find A-1 if A = 0 1 1 1 0 1 . Also, show that A-1= A - 3I 1 1 0 2 45. Find A-1 if A = 1 2 5 Hence ,find the following system of equations:

1 -1 -1 x+2y+5z=10, x-y-z+2=0, 2x+3y-z+11=0. 2 3 -1

46. If X = = cos A sin A , prove that Xn = cos nA sin nA , nN - sin A cos A - sin nA cos nA [CBSE 2004] 47. Using properties of determinants, prove that 1 1 1

r = (-)(-r)(r-) r r 48. Using matrix method solve the following system of linear equations: x + y -z = 1, 3x +y-2z= 3, x-y-z = -1 [ CBSE 2004] 1 2 2 49. If A = 2 1 2 , prove that A - 4A 5I = 0. [ CBSE 2004] 2 2 1

50. Using properties of dets/. prove that b+c c+a a+b c+a c+b b+c = (a+b+c) a+b b+c c+a (ab+ bc+ ca-a-b-c)

[ CBSE 2004]

51. If A = 1 2 , f(x) = x- 2x-3 , show that f(A)= 0. [ CBSE 05] 2 1 a+x a-x a-x 52. Using properties of determinants, solve for x: a-x a+x a-x = 0. a-x a-x a+x [ CBSE 05] 53. Using matrix method solve the following system of linear equations: x+y-z = 1, x-y z = -1, 3x+y-2z = 3. [ CBSE 05]


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54. If A= cos i sin then prove by principle of Mathematical induction that i sin cos , An = cos n i sin n i sin n cos n , nN. [ CBSE 05] 55. Using matrix method solve the following system of linear equations: x+2y+z = 7, x+3z = 11, 2x - 3y = 1. [ CBSE 05] 56. Find the value of x, if 1 2 3 1 [ 1 x 1] 4 5 6 -2 = 0.

3 2 5 3 [ CBSE 05] 57. Express the matrix A = 1 3 5 as the sum o a symmetric and the skew -6 8 3 symmetric matrix. -4 6 5 [ CBSE 2006] 58. Using properties of determinants, prove the following: 3a -a+b -a+c a-b 3b c-b = 3(a+b+c)(ab+bc+ca) [ CBSE 2006] a-c b-c 3c 59. Using matrices , solve the following system of equations 3x-y+z =5, 2x-2y+3z = 7, x+y-z = -1. [ CBSE 2006] 60. If A = 6 5 , show that A2 12A + I = 0 . Hence find A-1. [ CBSE 06] 7 6 61. If a ,b and c are in A.P. show the following: x+1 x+2 x+a x+2 x+3 x+b = 0 [ CBSE 06] x+3 x+4 x+ c 62. Using matrices , solve the following system of equations X+2y-3z = 6, 3x+2y-2z = 3, 2x-y+z = 2. [ CBSE 2006] 63. If A = 3 2 find the values of a and b such that A2 + Aa + b = 0. 1 1 , hence find A-1. [ CBSE 06]. 64. Using properties of determinats, prove the following: 1 1 1 a b c = (a-b)(b-c)(c-a) [ CBSE 06] bc ca ab 65. Using matrices, solve the following system of equations 2x -y + z= 0, x+ y - z = 6, 3x y 4z = 7. [ CBSE 06]


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66. Express the following matrices as the sum of a symmetric and a skew-symmetric matrices.[ CBSE D2006]

1 3 5 -6 8 3 -4 6 5

67. Using properties of determinants, prove the following: .[CBSED 06] a b c a-b b-c c-a = a3 + b3 + c3 -3abc b+c c+a a+b 68. If A = 3 -4 , find matrix B such that AB = I. [CBSE D06] -1 2 69. If A= cos i sin then prove by principle of Mathematical induction that i sin cos , An = cos n i sin n i sin n cos n , nN. [ CBSE D 06] 70. If A = 3 1 , verify that A2 5A + 7I = 0. Hence find A-1. -1 2 71. If A = 2 3 , verify that A 2 - 4A + I = 0 , hence find A-1. 1 2 72. Using properties of determinants, prove the following: ( b+c)2 a2 a2

b2 (c +a)2 b2 = 2abc( a+b+c)3

c2 c2 ( a+b)2

73. If A = 2 3 , prove that A3 - 4A2 + A = 0 . 1 2 74. Using matrices, solve the following system of linear equations: X+y+z = 4, 2x-y+z = -1, 2x+y-3z = - 9. [CBSE F 05] 75. If A = 3 1 , find x and y such that A2 + xI = y A, hence find A-1. 7 5 [ D 05] 76.If A = 2 -1 , B = 0 4 , find 3 A2 - 2B + I . 3 2 -1 7


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77. If A = 1 0 find k such that A2 8A + kI= 0. -1 7 78. If A = -1 2 , B = -2 -1 -4 , verify that ( AB)1 = B1 A1. 3 79. Using the properties of determinants, prove that a b c a2 b2 c2 = ( a+b+c)( a-b)(b-c)(c-a) b+c c+a a+b 80. Using matrix method solve the following system of equations for x, y and z. 2 3 + 3 = 10 ; 1 + 1 + 1 = 10 ; 3 1 + 2 = 13 ; x y z x y z x y z 81. Using properties of determinants, solve the following for x: x+4 2x 2x 2x x+4 2x = 0. 2x 2x x+4 82. Using properties of determinants, prove the following : a2 bc c 2 +ac a2 + ab b2 ca = 4 a2 b2 c2. ab b2 + bc c2

3 1 83. If A = , show that A2 5A + 7I = 0. Hence find A-1. [ CBSE 07] -1 2 84. Using properties of determinants, prove that a+ b b+c c+a a b c b +c c+a a+b = 2 b c a [CBSE 07] c+ a a+b b+c c a b 85. Using matrices, solve the following system of equations: x + 2y + z = 7, x + 3z = 11, 2x 3y = 1. [ CBSE 07]


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86. Using the properties of determinants, prove that x-3 x-4 x- x-2 x-3 x- = 0, where , and r are in A.P. [CBSE 07] x-1 x-2 x- r 87. Using matrices, solve the following system of equations: 4x - 5y - 11z = 12, x 3y + z =1, 2x + 3y 7z = 2. [CBSE 07] 88. If A = 3 -2 , find the value of so that A2 = A 2 I . Hence find A-1. 4 -2 [CBSE 07] 89. Using the properties of determinants, prove that x+4 2x 2x 2x x+4 2x = (5x + 4) (4 - x)2 [CBSE 07] 2x 2x x+4 90. Using elementary transformations, find the inverse of each of the matrices, if it exists

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