Class XII NCERT Maths Chapter 3 - Matrices
Class XII – NCERT – MathsChapter 3 - MatricesExercise 3.1Question 1: 25 19 7 5 In the matrix A 35 212 ,write: 2 3 1 5 17 (i) The order of the matrix(ii) The number of elements,(iii) Write the elements a13 , a21 , a33 , a24 , a23Solution 1:(i) In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, theorder of the matrix is 3 4 .(ii) Since the order of the matrix is 3 4 , there are 3 4 12 elements in it.(iii) a13 19, a21 35, a33 5, a24 12, a23 52Question 2:If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?Solution 2:We know that if a matrix is of the order m n , it has mn elements.Thus, to find all the possible orders of a matrix having 24 elements, we have to find all theordered pairs of natural numbers whose product is 24.The ordered pairs are: 1, 24 , 24,1 , 2,12 , 12, 2 , 3,8 , 8,3 , 4,6 , and 6, 4 Hence, the possible orders of a matrix having 24 elements are:1 24, 24 1, 2 12,12 2,3 8,8 3, 4 6 and 6 4 1,13 and 13,1 are the ordered pairs of natural numbers whose product is 13.Hence the possible orders of matrix having 13 elements are 1 13 and 13 1 .3.Matrices
Class XII – NCERT – MathsChapter 3 - MatricesQuestion 3:If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?Solution 3:We know that if a matrix is of the order m n , it has mn elements.Thus, to find all the possible orders of a matrix having 18 elements, we have to find all theordered pairs of natural numbers whose products is 18.The ordered pairs are: 1,18 , 18,1 , 2,9 , 9, 2 , 3,6 , and 6,3 Hence, the possible orders of a matrix having 18 elements are :1 18,18 1, 2 9,9 2,3 6 and 6 3 1,5 and 5,1 are the ordered pairs of natural numbers whose product is 5.Hence, the possible orders of a matrix having 5 elements are 1 5 and 5 1.Question 4:Construct a 3 x 4 matrix, whose elements are given by1(i) aij 3i j(ii) aij 2i j2Solution 4: a11 a12 a13In general, a 3 x 4 matrix is given by A a21 a22 a23 a31 a32 a331(i) Given : aij 3i j , i 1, 2, 3 and j 1, 2, 3, 42Thus, we have1112 a11 3 1 1 3 1 2 12222 a21 1115 3 2 1 6 1 5 2222 a31 111 3 3 1 9 1 8 4222 a12 1111 3 1 2 3 2 1 22223.Matricesa14 a24 a34
Class XII – NCERT – MathsChapter 3 - Matrices a22 1114 3 2 2 6 2 4 22222 a32 1117 3 3 2 9 2 7 2222 a13 11 3 1 3 3 3 022 a23 1113 3 2 3 6 3 3 2222 a33 1116 3 3 3 9 3 6 32222 a14 1111 3 1 4 3 4 1 2222 a24 1112 3 2 4 6 4 2 12222 a34 1115 3 3 4 9 4 5 2222 1 5Therefore, the required matrix is A 2 4 120232723(ii) Given : aij 2i j , i 1, 2,3 and j 1, 2,3, 4Thus, we have a11 2 1 1 2 1 13.Matrices1 2 1 5 2
Chapter 3 - MatricesClass XII – NCERT – Maths a21 2 2 1 4 1 3 a31 2 3 1 6 1 5 a12 2 1 2 2 2 0 a22 2 2 2 4 2 2 a32 2 3 2 6 2 4 a13 2 1 3 2 3 1 a23 2 2 3 4 3 1 a33 2 3 3 6 3 3 a14 2 1 4 2 4 2 a24 2 2 4 4 4 0 a34 2 3 4 6 4 2 1 0 1 2 Therefore, the required matrix is A 3 2 1 0 5 4 3 2 Question 5:Find the value of x, y and z from the following equation: 4 3 y z (i) x 5 1 5 x y(ii) 5 z2 6 2 xy 5 8 x y z 9 (iii) x z 5 y z 7 Solution 5: 4 3 y z (i) Given : x 5 1 5 As the given matrices are equal, their corresponding elements are also equal.3.Matrices
Chapter 3 - MatricesClass XII – NCERT – MathsComparing the corresponding elements, we get:x 1, y 4 and z 3 x y(ii) 5 z2 6 2 xy 5 8 As the given matrices are equal, their corresponding elements are also equal.Comparing the corresponding elements, we get:x y 6, xy 8, 5 z 5Now, 5 z 5 z 0Using x y x y 4 xy , we get22 x y 36 32 42 x y 2 When x y 2 and x y 6 ,we get x 4 and y 2 When x y 2 and x y 6 we get x 2 and y 4 x 4, y 2, and z 0 or x 2 , y 4 , and z 0 x y z 9 (iii) x z 5 y z 7 As the two matrices are equal, their corresponding elements are also equal.Comparing the corresponding elements, we get:x y z 9.(1)x z 5.(2)y z 7.(3)From (1) and (2), we have:y 5 9 y 4From (3), we have:4 z 7 z 33.Matrices
Class XII – NCERT – MathsChapter 3 - Matrices x z 5 x 2 x 2, y 4, and z 3Question 6:Find the value of a, b , c , and d from the equation: a b 2 a c 1 5 2a b 3c d 0 13 Solution 6: a b 2 a c 1 5 2a b 3c d 0 13 As the two matrices are equal, their corresponding elements are also equal.Comparing the corresponding elements, we get:a b 12a b 02a c 53c d 13.(1).(2).(3).(4)From (2), we have:b 2aThen, from (1), we have:a 2a 1 a 1 b 2Now, from (3), we have:2 1 c 5 c 3From (4) we have:3.Matrices
Chapter 3 - MatricesClass XII – NCERT – Maths3 3 d 13 9 d 13 d 4 a 1, b 2, c 3, and d 4Question 7:A ai j is a square matrix, ifm nA. m nB. m nC. m nD. None of theseSolution 7:The correct answer is C.It is known that a given matrix is said to be a square matrix if the number of rows is equal tothe number of columns.Therefore, A ai j is a square matrix, if m n.m nQuestion 8:Which of the given values of x and y make the following pair of matrices equal5 0 3 x 7 y 1 2 3 x 8 y 2 4 1, y 73B. Not possible to find 2C. y 7, x 3 1 2D. x , y 33A.x Solution 8:The Correct answer is B.3.Matrices
Class XII – NCERT – Maths5 0 3 x 7It is given that y 1 2 3 x 8Chapter 3 - Matricesy 2 4 Equating the corresponding elements, we get:73x 7 0 x 35 y 2 y 7y 1 8 y 722 3x 4 x 3We find that on comparing the corresponding elements of the two matrices, we get twodifferent values of x, which is not possible.Hence, it is not possible to find the values of x and y for which the given matrices are equal.Question 9:The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is:A.B.C.D.271881512Solution 9:The correct answer is D.The given matrix of the order 3 x 3 has 9 elements and each of these elements can be either0 or 1.Now, each of the 9 elements can be filled in two possible ways.Therefore, by the multiplication principle, the required number of possible matrices is29 512Exercise 3.2Question 1: 2 4 1 3 2 5 ,B ,C Let A 3 2 2 5 3 4 3.Matrices
Chapter 3 - MatricesClass XII – NCERT – MathsFind each of the following(i) A B(iii) 3A C(v) BA(ii) A B(iv) ABSolution 1: 2 4 1 3 2 1 4 3 3(i) A B 3 2 2 5 3 2 2 5 17 7 2 4 1 3 2 1 4 3 1(ii) A B 3 2 2 5 3 2 2 5 51 3 2 4 2 5 (iii) 3 A C 3 3 2 3 4 3 2 3 4 2 5 3 3 3 2 3 4 6 12 2 5 9 6 3 4 6 2 12 5 9 3 6 4 8 7 6 2 (iv) Matrix A has 2 columns. This number is equal to the number of rows in matrix B.Therefore, AB is defined as:3.Matrices
Chapter 3 - MatricesClass XII – NCERT – Maths 2 4 1 3 AB 3 2 2 5 2 1 4 2 3 1 2 2 2 3 4 5 3 3 2 5 2 8 6 20 3 4 9 10 6 26 1 19 (v) matrix B has 2 columns. This number is equal to the number of rows in matrix A.Therefore, BA is defined as:1 4 3 2 1 3 2 4 1 2 3 3 BA 2 5 3 2 2 2 5 3 2 4 5 2 2 9 4 154 6 11 10 8 10 11 2 Question 2:Compute the following: a b a b (i) b a b a a 2 b2 b2 c 2 2ab(ii) 2 2 2 2 a c a b 2ac2bc 2ab 1 4 6 12 7 6 (iii) 8 5 16 8 0 5 2 8 5 3 2 4 cos2 x sin 2 x sin 2 x cos2 x (iv) 2 22 2 sin x cos x cos x sin x 3.Matrices
Chapter 3 - MatricesClass XII – NCERT – MathsSolution 2: a b a(i) b a bb a a b b 2a 2b a b b a a 0 2a a 2 b2 b2 c 2 2ab(ii) 2 2 2 2 a c a b 2ac a 2 b 2 2ab 2 2 a c 2ac a b 2 2 a c 2bc 2ab b 2 c 2 2bc a 2 b 2 2ab b c 2 a b 2 1 4 6 12 7 6 (iii) 8 5 16 8 0 5 2 8 5 3 2 4 1 12 4 7 6 6 8 8 5 0 16 5 2 3 8 2 5 4 11 11 0 16 5 21 5 10 9 cos2 x sin 2 x sin 2 x cos2 x (iv) 2 22 2 sin x cos x cos x sin x cos 2 x sin 2 x sin 2 x cos 2 x 2222 sin x cos x cos x sin x 1 1 sin 2 x cos 2 x 1 1 1 3.Matrices
Class XII – NCERT – MathsChapter 3 - MatricesQuestion 3:Compute the indicated products a b a b (i) b a b a 1 (ii) 2 2 3 4 3 1 2 1 2 3 (iii) 2 3 2 3 1 2 3 4 1 3 5 (iv) 3 4 5 0 2 4 4 5 6 3 0 5 2 1 1 0 1 (v) 3 2 1 2 1 1 1 2 3 3 1 3 1 0 (vi) 1 0 2 3 1 Solution 3: a b a b (i) b a b a a a b b b a a b a 2 b2 ab ab3.Matricesa b b a b b a a ab ab a 2 b 2 b2 a 2 00 a b2 2
Class XII – NCERT – MathsChapter 3 - Matrices 1 2 1 3 1 4 2 3 4 1 (ii) 2 2 3 4 2 2 2 3 2 4 4 6 8 3 2 3 3 3 4 6 9 12 3 1 2 1 2 3 (iii) 2 3 2 3 1 1 1 2 2 1 2 2 3 1 3 2 1 2 1 3 2 2 2 3 3 2 3 3 1 1 4 2 6 3 2 3 4 1 2 6 4 9 6 3 8 13 9 2 3 4 1 3 5 (iV) 3 4 5 0 2 4 4 5 6 3 0 5 2 1 3 0 4 3 2 3 3 2 4 0 2 5 3 4 4 5 3 1 4 0 5 3 3 3 4 2 5 0 3 5 4 4 5 5 4 1 5 0 6 3 4 3 5 2 6 0 4 5 5 4 6 5 2 0 12 6 6 0 10 12 20 14 0 42 3 0 15 9 8 0 15 16 25 18 1 56 4 0 18 12 10 0 20 20 30 22 2 70 2 1 1 0 1 (v) 3 2 1 2 1 1 1 3.Matrices
Class XII – NCERT – MathsChapter 3 - Matrices 2 1 1 1 2 0 1 2 2 1 1 1 3 1 2 1 3 0 2 2 3 1 2 1 1 1 1 1 1 0 1 2 1 1 1 2 1 0 2 2 1 1 2 3 3 2 0 4 3 2 1 4 5 1 1 0 2 1 1 2 2 0 2 3 3 1 3 1 0 (vi) 1 0 2 3 1 3 3 1 0 3 1 3 2 1 1 3 3 1 2 0 1 2 3 1 3 0 0 2 1 6 1 9 9 0 3 14 6 2 0 6 3 0 2 4 5 Question 4: 1 2 3 3 1 2 4 1 2 If A 5 0 2 , B 4 2 5 , and C 0 3 2 , then 1 1 1 2 0 3 1 2 3 Compute A B and B C . Also, verify that A B C A B CSolution 4: 1 2 3 3 1 2 A B 5 0 2 4 2 5 1 1 1 2 0 3 1 3 2 1 3 2 4 1 1 5 4 0 2 2 5 9 2 7 1 2 1 0 1 3 3 1 4 3.Matrices
Chapter 3 - MatricesClass XII – NCERT – Maths 3 1 2 4 1 2 B C 4 2 5 0 3 2 2 0 3 1 2 3 1 2 3 1 2 0 A B C 5 0 2 4 1 3 1 1 1 1 2 0 1 1 2 2 3 0 0 0 3 5 40 1 2 3 9 1 5 1 1 1 21 0 2 1 1 4 1 1 4 1 2 A B C 9 2 7 0 3 2 3 1 4 1 2 3 1 1 1 2 0 0 3 4 4 9 02 37 2 9 1 5 3 1 1 2 4 3 2 1 1 Hence, we have verified that A B C A B C.Question 5: 2 3 1 1 2If A 3 3 7 2 3Solution 5:3.Matrices5 3 4 3 2 3 and 2 5 1B 5 7 5352565 1 4 then compute 3 A 5B5 2 5
Chapter 3 - MatricesClass XII – NCERT – Maths 2 3 13 A 5B 3 3 7 312325 3 4 53 2 3 2 5 1 5 7 5352565 1 4 5 2 5 2 3 5 2 3 5 0 0 0 1 2 4 1 2 4 0 0 0 7 6 2 7 6 2 0 0 0 Question 6: cos Slimplify cos sin Solution 6: cos cos sin sin sin sin cos cos sin sin sin cos cos cos2 sin cos 1 0 0 1 cos sin cos sin sin 2 cos 2 sin cos cos 2 sin 2 sin cos sin cos sin cos sin 2 cos sin sin cos cos 2 sin 2 cos 2 sin 2 1 Question 7:Find X and Y , if 7 0 3 0 and X Y (i) X Y 2 5 0 3 3.Matrices cos sin
Chapter 3 - MatricesClass XII – NCERT – Maths 2 3 2 2 and 3 X 2Y (ii) 2 X 3Y 4 0 1 5 Solution 7: 7(i) X Y 2 3X Y 00 5 .(1)0 3 .(2)Adding equations (1) and (2), we get: 7 0 3 0 7 3 0 0 10 0 2X 2 5 0 3 2 0 5 3 2 8 X 1 10 0 5 0 2 2 8 1 4 7 0 Now, X Y 2 5 5 10 7 Y 4 20 5 7 0 5 0 Y 2 5 1 4 7 5 0 0 Y 2 1 5 4 2 Y 10 1 2 3 (ii) 2 X 3Y 4 0 3.Matrices.(3)
Chapter 3 - MatricesClass XII – NCERT – Maths 2 2 3 X 2Y .(4) 1 5 Multiplying equation (3) with (2), we get 22 2 X 3Y 2 43 0 4 6 4 X 6Y .(5) 8 0 Multiplying equation (4) with (3), we get 2 2 3 3 X 2Y 3 1 5 6 6 9 X 6Y 3 15 .(6)From (5) and (6), we have 4X 4 6 6 6Y 9 X 6Y 8 0 3 4 6 6 6 2 12 5X 83 015 11 15 1 2 X 5 11 212 5 15 11 5 2 3 2 X 3Y 4 0 Now,3.Matrices 12 5 3 6 15
Chapter 3 - MatricesClass XII – NCERT – Maths 2 5 2 11 5 4 5 22 5 2 3Y 4 12 2 3 5 3Y 4 0 3 24 25 3Y 46 43 5 0 22 54 2 5 3Y 4 225 61 5 Y 3 42 53 0 24 5 6 624 3 55 420 6 5 39 5 6 39 2 13 5 5 5 14 6 2 5 Question 8: 3Find X , if Y 1Solution 8: 12X Y 3 3 2X 13.Matrices2 1 0 and 2 X Y 4 3 2 0 2 2 1 4 30 2
Chapter 3 - MatricesClass XII – NCERT – Maths 1 2X 3 2 2X 4 X 0 3 2 1 2 2 1 22 42 1 3 0 2 4 3 1 2 4 2 1 2 2 1 1 Question 9: 1Find X and Y , if 2 0Solution 9:3 y 12 x 1 00 5 2 1 2 6 y 0 2 x 1 2 y 13 y x 10 5 2 16 8 6 8 0 5 6 2 1 8 6 5 2 x 2 16 8 Comparing the corresponding elements of these two matrices, we have:2 y 5 y 32x 2 8 x 3 x 3 and y 3Question 10:Solve the equation for x, y, z and t if x2 y3.Matricesz 1 3 t 0 1 3 3 2 45 6
Chapter 3 - MatricesClass XII – NCERT – MathsSolution 10: x z 12 3 y t 0 2 x 2 z 3 2 y 2t 0 1 3 3 2 45 6 3 9 15 6 12 18 2 x 3 2 z 3 9 15 2t 6 12 18 2yComparing the corresponding elements of these two matrices, we get:2x 3 9 2x 6 x 32 y 12 y 62 z 3 15 2 z 18 z 92t 6 18 2t 12 t 6 x 3, y 6, z 9, and t 6Question 11: 2 1 10 If x y , find values of x and y. 3 1 5 3.Matrices
Class XII – NCERT – MathsChapter 3 - MatricesSolution 11: 2 1 10 x y 3 1 5 2 x y 10 3x y 5 2 x y 10 3x y 5 Comparing the corresponding elements of these two matrices, we get:2 x y 10and 3x y 5Adding these two equations, we have:5x 15 x 3Now, 3x y 5 y 5 3x y 5 9 4 x 3 and y 4Question 12: x y x 6 4 x y Given 3 , find the values of x , y, z and w. z w 1 2 w z w 3 Solution 12: x y x 6 4 x y 3 z w 1 2w z w 3 6 x y 3x 3 y x 4 2 w 3 3z 3w 1 z wComparing the corresponding elements of these two matrices, we get:3.Matrices
Class XII – NCERT – MathsChapter 3 - Matrices3x x 4 2x 4 x 23x 6 x y 2y 6 x 6 2 8 y 43w 2 w 3 w 33z 1 z w 2z 1 w 1 3 2 z 1 x 2, y 4, z 1, and w 3Question 13: cos x sin x 0 If F x sin x cos x 0 , show that F x F y F x y . 001 Solution 13: cos x sin x 0 cos y sin y 0 F x sin x cos x 0 , F y sin y cos y 0 0 001 01 cos x y sin x y 0 R.H.S : F x y sin x y cos x y 0 001 L.H.S : F x F y cos x sin x 0 cos y sin y 0 sin x cos x 0 sin y cos y 0 01 001 03.Matrices
Chapter 3 - MatricesClass XII – NCERT – Maths cos x cos y sin x sin y 0 sin x cos y cos x sin y 0 0 cos x y sin x y sin x y cos x y 00 F x y cos x sin y sin x cos y 0 0 sin x sin y cos x cos y 0 0 00 0 0 1 F x F y F x y Question 14:Show that 5 1 2(i) 6 7 31 2 4 31 5 1 4 6 7 1 2 3 1 1 0 1 1 0 1 2 3 (ii) 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 2 3 4 2 3 4 1 1 0 Solution 14: 5 1 2(i) 6 7 31 4 5 2 1 3 5 1 1 4 6 2 7 3 6 1 7 4 10 3 5 4 7 12 21 6 28 33 2 3 1 54 61 34 1 7 2 5 1 6 2 1 1 7 3 5 4 6 3 1 4 7 3.Matrices
Chapter 3 - MatricesClass XII – NCERT – Maths 10 6 2 7 16 5 15 24 3 28 39 25 5 1 2 6 7 31 2 1 5 4 3 4 6 1 7 1 2 3 1 1 0 (ii) 0 1 0 0 1 1 1 1 0 2 3 4 1 1 2 0 3 2 1 1 2 1 3 3 1 0 2 1 3 4 0 1 1 0 0 2 0 1 1 1 0 3 0 0 1 1 0 4 1 1 1 0 0 2 1 1 1 1 0 3 1 0 1 1 0 4 5 8 14 0 1 1 1 0 1 1 1 0 1 2 3 0 1 1 0 1 0 2 3 4 1 1 0 1 1 1 0 0 1 1 2 1 1 0 1 1 3 1 0 0 0 0 1 1 0 1 1 0 2 1 1 1 1 0 3 1 0 1 0 2 2 3 1 4 1 2 3 3 0 4 0 2 1 3 0 4 1 1 1 3 1 0 0 6 11 6 1 2 3 1 1 0 1 1 0 1 2 3 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 2 3 4 2 3 4 1 1 0 3.Matrices
Class XII – NCERT – MathsChapter 3 - MatricesQuestion 15: 2 0 1 Find A 5 A 6 I if A 2 1 3 1 1 0 2Solution 15:We have A2 A x A 2 0 1 2 0 1 A AA 2 1 3 2 1 3 1 1 0 1 1 0 22 0 0 1 1 1 2 1 0 3 1 0 2 2 0 2 1 1 2 0 1 1 3 1 2 1 1 3 3 0 2 2 1 2 3 1 1 2 1 2 0 1 1 0 1 1 0 1 1 1 1 3 0 0 4 0 1 0 0 1 2 0 0 4 2 3 0 1 3 2 3 0 2 2 0 0 1 0 1 3 0 5 1 2 9 2 5 0 1 2 Substituting the matrices in the given equation : A2 5 A 6 I 5 1 2 2 0 1 1 0 0 9 2 5 5 2 1 3 6 0 1 0 0 1 2 1 1 0 0 0 1 5 1 2 10 0 5 6 0 0 9 2 5 10 5 15 0 6 0 0 1 2 5 5 0 0 0 6 3.Matrices
Class XII – NCERT – MathsChapter 3 - Matrices 5 10 1 0 2 5 6 0 0 9 10 2 5 5 15 0 6 0 0 5 1 5 2 0 0 0 6 5 1 3 6 0 0 1 7 10 0 6 0 5 4 2
Class XII – NCERT – Maths Chapter 3 - Matrices 3.Matrices . Exercise 3.1 . Question 1: In the matrix 2 5 19 7 5 35 2 12 2 3 1 5 17. A . As the given matrices are equal, their corresponding elements are also equal. Class XII – NCERT – Maths . Chapter 3 - Matrices . 3.Matrices . Comparing the corresponding elements, we get: .
Number of Clusters XII-9. C) Overlap XII-10. D) An Example XII-10. 5. Implementation XII-13. A) Storage Management XII-14. 6. Results XII-14. A) Clustering Costs XII-15. B) Effect of Document Ordering XII-19. Cl. Search Results on Clustered ADI Collection . XII-20. D) Search Results of Clustered Cranfield Collection. XII-31. 7. Conslusions XII .
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in Prep Course Lesson Book A of ALFRED'S BASIC PIANO LIBRARY. It gives the teacher considerable flexibility and is intended in no way to restrict the lesson procedures. FORM OF GUIDE The Guide is presented basically in outline form. The relative importance of each activity is reflected in the words used to introduce each portion of the outline, such as EMPHASIZE, SUGGESTION, IMPORTANT .
