Matrices and determinants_01

Matrices and determinants_01

• 1.
  23 XII –Maths CHAPTER 3 & 4 MATRICES AND DETERMINANTS POINTS TO REMEMBER  Matrix : A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements of the matrix.  Order of Matrix : A matrix having ‘m’ rows and ‘n’ columns is called the matrix of order mxn.  Square Matrix : An mxn matrix is said to be a square matrix of order n if m = n.  Column Matrix : A matrix having only one column is called a column matrix i.e. A = [aij]mx1 is a column matrix of order mx1.  Row Matrix : A matrix having only one row is called a row matrix i.e.  1  xn B bij is a row matrix of order 1xn.  Zero Matrix : A matrix having all the elements zero is called zero matrix or null matrix.  Diagonal Matrix : A square matrix is called a diagonal matrix if all its non diagonal elements are zero.  Scalar Matrix : A diagonal matrix in which all diagonal elements are equal is called a scalar matrix.  Identity Matrix : A scalar matrix in which each diagonal element is 1, is called an identity matrix or a unit matrix. It is denoted by I.  I = [eij]n × n where,     1 if 0 ifij i j e i j
• 2.
  XII – Maths24  Transpose of a Matrix : If A = [ai j ]m × n be an m × n matrix then the matrix obtained by interchanging the rows and columns of A is called the transpose of the matrix. Transpose of A is denoted by A´ or AT. Properties of the transpose of a matrix. (i) (A´)´ = A (ii) (A + B)´ = A´ + B´ (iii) (kA)´ = kA´, k is a scalar (iv) (AB)´ = B´A´  Symmetrix Matrix : A square matrix A = [aij ] is symmetrix if aij = aji  i, j. Also a square matrix A is symmetrix if A´ = A.  Skew Symmetrix Matrix : A square matrix A = [aij] is skew-symmetrix, if aij = – aji  i, j. Also a square matrix A is skew - symmetrix, if A´ = – A.  Determinant : To every square matrix A = [aij] of order n × n, we can associate a number (real or complex) called determinant of A. It is denoted by det A or |A|. Properties (i) |AB| = |A| |B| (ii) |kA|n × n = kn |A|n × n where k is a scalar. Area of triangles with vertices (x1, y1), (x2, y2) and (x3, y3) is given by   1 1 2 2 3 3 1 1 1 2 1 x y x y x y The points (x1, y1), (x2, y2), (x3, y3) are collinear 1 1 2 2 3 3 1 1 0 1 x y x y x y    Adjoint of a Square Matrix A is the transpose of the matrix whose elements have been replaced by their cofactors and is denoted as adj A. Let A = [aij]n × n adj A = [Aji]n × n
• 3.
  25 XII –Maths Properties (i) A(adj A) = (adj A) A = |A| I (ii) If A is a square matrix of order n then |adj A| = |A|n–1 (iii) adj (AB) = (adj B) (adj A). [Note : Correctness of adj A can be checked by using A.(adj A) = (adj A) . A = |A| I ] Singular Matrix : A square matrix is called singular if |A| = 0, otherwise it will be called a non-singular matrix. Inverse of a Matrix : A square matrix whose inverse exists, is called invertible matrix. Inverse of only a non-singular matrix exists. Inverse of a matrix A is denoted by A–1 and is given by   1 1 . .A adj A A Properties (i) AA–1 = A–1A = I (ii) (A–1)–1 = A (iii) (AB)–1 = B–1A–1 (iv) (AT)–1 = (A–1)T  Solution of system of equations using matrix : If AX = B is a matrix equation then its solution is X = A–1B. (i) If |A|  0, system is consistent and has a unique solution. (ii) If |A| = 0 and (adj A) B  0 then system is inconsistent and has no solution. (iii) If |A| = 0 and (adj A) B = 0 then system is either consistent and has infinitely many solutions or system is inconsistent and has no solution.
• 4.
  XII – Maths26 VERY SHORT ANSWER TYPE QUESTIONS (1 Mark) 1. If 3 4 5 4 , 4 3 9 x y x y             find x and y. 2. If 0 0 and , find . 0 0 i i A B AB i i             3. Find the value of a23 + a32 in the matrix A = [aij]3 × 3 where 2 – if 2 3 if ij i j i j a i j i j        . 4. If B be a 4 × 5 type matrix, then what is the number of elements in the third column. 5. If 5 2 3 6 and find 3 2 . 0 9 0 1 A B A B              6. If                 2 3 1 0 and find . 7 5 2 6 ´A B A B 7. If A = [1 0 4] and 2 5 find . 6 B AB          8. If 4 2 2 3 1 x A x x        is symmetric matrix, then find x. 9. For what value of x the matrix 0 2 3 2 0 4 3 4 5x          is skew symmetrix matrix. 10. If 2 3 1 0 A P Q         where P is symmetric and Q is skew-symmetric matrix, then find the matrix Q.
• 5.
  27 XII –Maths 11. Find the value of a ib c id c id a ib      12. If 2 5 3 0, find . 5 2 9 x x x    13. For what value of k, the matrix 2 3 4 k      has no inverse. 14. If sin 30 cos 30 , sin 60 cos 60 A          what is |A|. 15. Find the cofactor of a12 in 2 3 5 6 0 4 . 1 5 7   16. Find the minor of a23 in 1 3 2 4 5 6 . 3 5 2   17. Find the value of P, such that the matrix 1 2 4 P       is singular. 18. Find the value of x such that the points (0, 2), (1, x) and (3, 1) are collinear. 19. Area of a triangle with vertices (k, 0), (1, 1) and (0, 3) is 5 unit. Find the value (s) of k. 20. If A is a square matrix of order 3 and |A| = – 2, find the value of |–3A|. 21. If A = 2B where A and B are square matrices of order 3 × 3 and |B| = 5, what is |A|? 22. What is the number of all possible matrices of order 2 × 3 with each entry 0, 1 or 2. 23. Find the area of the triangle with vertices (0, 0), (6, 0) and (4, 3). 24. If 2 4 6 3 , find . 1 2 1 x x x   
• 6.
  XII – Maths28 25. If , 1 1 1            x y y z z x A z x y write the value of det A. 26. If 11 12 21 22 a a A a a        such that |A| = – 15, find a11 C21 + a12C22 where Cij is cofactors of aij in A = [aij]. 27. If A is a non-singular matrix of order 3 and |A| = – 3 find |adj A|. 28. If          5 3 find 6 8 A adj A 29. Given a square matrix A of order 3 × 3 such that |A| = 12 find the value of |A adj A|. 30. If A is a square matrix of order 3 such that |adj A| = 8 find |A|. 31. Let A be a non-singular square matrix of order 3 × 3 find |adj A| if |A| = 10. 32. If   112 1 find . 3 4 A A        33. If   3 1 2 3 and 4 0 A B            find |AB|. SHORT ANSWER TYPE QUESTIONS (4 MARKS) 34. Find x, y, z and w if 2 –1 5 . 2 3 0 13 x y x z x y x w              35. Construct a 3 × 3 matrix A = [aij] whose elements are given by aij = 1 if 2 if 2 i j i j i j i j       
• 7.
  29 XII –Maths 36. Find A and B if 2A + 3B = 1 2 3 3 0 1 and 2 2 0 1 1 6 2 A B              . 37. If   1 2 and 2 1 4 , 3 A B             verify that (AB)´ = B´A´. 38. Express the matrix 3 3 1 2 2 1 4 5 2 P Q             where P is a symmetric and Q is a skew-symmetric matrix. 39. If A = cos sin , sin cos         then prove that cos sin sin cos n n n A n n          where n is a natural number. 40. Let 2 1 5 2 2 5 , , , 3 4 7 4 3 8 A B C                     find a matrix D such that CD – AB = O. 41. Find the value of x such that   1 3 2 1 1 1 2 5 1 2 0 15 3 2 x x                    42. Prove that the product of the matrices 22 22 cos cos sincos cos sin and cos sin sincos sin sin                       is the null matrix, when  and  differ by an odd multiple of . 2  43. If 5 3 12 7 A        show that A2 – 12A – I = 0. Hence find A–1.
• 8.
  XII – Maths30 44. If 2 3 4 7 A        find f(A) where f(x) = x2 – 5x – 2. 45. If 4 3 , 2 5 A        find x and y such that A2 – xA + yI = 0. 46. Find the matrix X so that               1 2 3 7 8 9 4 5 6 2 4 6 X . 47. If 2 3 1 2 and 1 4 1 3 A B              then show that (AB)–1 = B–1A–1. 48. Test the consistency of the following system of equations by matrix method : 3x – y = 5; 6x – 2y = 3 49. Using elementary row transformations, find the inverse of the matrix 6 3 2 1 A       , if possible. 50. By using elementary column transformation, find the inverse of 3 1 . 5 2 A        51. If cos sin sin cos A          and A + A´ = I, then find the general value of . Using properties of determinants, prove the following : Q 52 to Q 59. 52.  3 2 2 2 2 2 2 a b c a a b b c a b a b c c c c a b          53. 2 3 2 3 4 2 0 if , , are in . . 4 5 2 x x x a x x x b a b c A P x x x c           54.       sin cos sin sin cos sin 0 sin cos sin                
• 9.
  31 XII –Maths 55. 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 .     b c a a b c a b a b c c c a b 56. 2 . b c c a a b a b c q r r p p q p q r y z z x x y x y z           57. 2 2 2 2 2 2 2 2 2 4 . a bc ac c a ab b ac a b c ab b bc c     58.  2 . x a b c a x b c x x a b c a b x c        59. Show that :        2 2 2 . x y z x y z y z z x x y yz zx xy yz zx xy       60. (i) If the points (a, b) (a´, b´) and (a – a´, b – b´) are collinear. Show that ab´ = a´b. (ii) If 2 5 4 3 and verity that . 2 1 2 5 A B AB A B               61. Given 0 1 0 1 2 and 1 0 . 2 2 0 1 1 A B               Find the product AB and also find (AB)–1. 62. Solve the following equation for x. 0. a x a x a x a x a x a x a x a x a x          
• 10.
  XII – Maths32 63. If 2 2 0 tan and is the identity tan 0           A I matrix of order 2, show that,   cos sin sin cos            A AI I 64. Use matrix method to solve the following system of equations : 5x – 7y = 2, 7x – 5y = 3. LONG ANSWER TYPE QUESTIONS (6 MARKS) 65. Obtain the inverse of the following matrix using elementary row operations 0 1 2 1 2 3 . 3 1 1 A          66. Use product 1 1 2 2 0 1 0 2 3 9 2 3 3 2 4 6 1 2                      to solve the system of equations x – y + 2z = 1, 2y – 3z = 1, 3x – 2y + 4z = 2. 67. Solve the following system of equations by matrix method, where x  0, y  0, z  0 2 3 3 1 1 1 3 1 2 10, 10, 13. x y z x y z x y z          68. F ind A–1, where 1 2 3 2 3 2 3 3 –4 A          , hence solve the system of linear equations : x + 2y – 3z = – 4 2x + 3y + 2z = 2 3x – 3y – 4z = 11
• 11.
  33 XII –Maths 69. The sum of three numbers is 2. If we subtract the second number from twice the first number, we get 3. By adding double the second number and the third number we get 0. Represent it algebraically and find the numbers using matrix method. 70. Compute the inverse of the matrix. 3 1 1 15 6 5 5 2 5 A           and verify that A–1 A = I3. 71. If the matrix –1 1 1 2 1 2 0 0 2 3 and 0 3 –1 , 3 2 4 1 0 2 A B                     then compute (AB)–1. 72. Using matrix method, solve the following system of linear equations : 2x – y = 4, 2y + z = 5, z + 2x = 7. 73. Find 1 0 1 1 if 1 0 1 . 1 1 0 A A          Also show that 2 1 3 . 2 A I A   74. Find the inverse of the matrix 1 2 2 1 3 0 0 2 1 A          by using elementary column transformations. 75. Let 2 3 1 2 A       and f(x) = x2 – 4x + 7. Show that f (A) = 0. Use this result to find A5. 76. If cos sin 0 sin cos 0 , 0 0 1 A             verify that A . (adj A) = (adj A) . A = |A| I3.
• 12.
  XII – Maths34 77. For the matrix 2 1 1 1 2 1 , 1 1 2 A           verify that A3 – 6A2 + 9A – 4I = 0, hence find A–1. 78. Find the matrix X for which 3 2 1 1 2 1 . . 7 5 2 1 0 4 X                   79. By using properties of determinants prove the following :   2 2 32 2 2 2 2 2 1 2 2 2 1 2 1 . 2 2 1 a b ab b ab a b a a b b a a b            80.         2 32 2 2 . y z xy zx xy x z yz xyz x y z xz yz x y       81. 3 2 3 2 4 3 2 . 3 6 3 10 6 3 a a b a b c a a b a b c a a a b a b c           82. If x, y, z are different and 2 3 2 3 2 3 1 1 0. 1 x x x y y y z z z     Show that xyz = – 1. 83. If x, y, z are the 10th, 13th and 15th terms of a G.P. find the value of log 10 1 log 13 1 . log 15 1 x y z  
• 13.
  35 XII –Maths 84. Using the properties of determinants, show that : 1 1 1 1 1 1 1 1 1 1 1 1 1                a b abc abc bc ca ab a b c c 85. Using properties of determinants prove that   2 2 32 2 2 2 bc b bc c bc a ac ac c ac ab bc ca a ab b ab ab             86. If 3 2 1 4 1 2 , 7 3 3          A find A–1 and hence solve the system of equations 3x + 4y + 7z = 14, 2x – y + 3z = 4, x + 2y – 3z = 0. ANSWERS 1. x = 2, y = 7 2. 0 1 1 0       3. 11. 4. 4 5. 9 6 0 29       . 6. 3 5 3 1       . 7. AB = [26]. 8. x = 5 9. x = – 5 10. 0 1 . 1 0      11. a2 + b2 + c2 + d2. 12. x = – 13 13. 3 2 k  14. |A| = 1. 15. 46 16. –4
• 14.
  XII – Maths36 17. P = – 8 18. 5 . 3 x  19. 10 . 3 k  20. 54. 21. 40. 22. 729 23. 9 sq. units 24. x = ± 2 25. 0 26. 0 27. 9 28. 8 3 . 6 5      29. 1728 30. |A| = 9 31. 100 32. 11 33. |AB| = – 11 34. x = 1, y = 2, z = 3, w = 4 35.          3 3 2 5 2 4 5 2 . 5 6 7 36. 11 9 9 5 2 1 7 7 7 7 7 7 , 1 18 4 4 12 5 7 7 7 7 7 7 A B                            40. 191 110 . 77 44 D         41. x = – 2 or – 14 43. 1 7 3 . 12 5 A       44. f(A) = 0 45. x = 9, y = 14 46. 1 2 . 2 0 x       
• 15.
  37 XII –Maths 48. Inconsistent 49. Inverse does not exist. 50. –1 2 1 . 5 3 A       51. 2 , 3 n n z       61.                –11 2 2 21 , . 2 2 2 16 AB AB 62 0, 3a 64. 11 1 , . 24 24 x y  65. –1 1 1 1 2 2 2 4 3 1 5 3 1 2 2 2 A                  . 66. x = 0, y = 5, z = 3 67. 1 1 1 , , 2 3 5 x y z   68. 1 6 17 13 1 14 5 8 67 15 9 1 A            69. x = 1, y = – 2, z = 2 70. 1 2 0 1 5 1 0 0 1 3 A          71.   1 16 12 1 1 21 11 7 19 10 2 3 AB           . 72. x = 3, y = 2, z = 1. 73. 1 1 1 1 1 1 1 1 2 1 1 1 A          . 74. 1 3 2 6 1 1 2 2 2 5 A         
• 16.
  XII – Maths38 75. 5 118 93 . 31 118 A        77. 1 3 1 1 1 1 3 1 . 4 1 1 3 A           78. 16 3 . 24 5       X 83. 0 86. x = 1, y = 1, z = 1.