Matrices PDF

56 MATHEMATICS Chapter 3 MATRICES he ™ The essence of Mathematics lies in its freedom. — CANTOR...

56 MATHEMATICS Chapter 3 MATRICES he ™ The essence of Mathematics lies in its freedom. — CANTOR ™ 3.1 Introduction is The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies bl our work to a great extent when compared with other straight forward methods. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrices are not only used as a pu representation of the coefficients in system of linear equations, but utility of matrices far exceeds that use. Matrix notation and operations are used in electronic spreadsheet be T programs for personal computer, which in turn is used in different areas of business re and science like budgeting, sales projection, cost estimation, analysing the results of an o R experiment etc. Also, many physical operations such as magnification, rotation and reflection through a plane can be represented mathematically by matrices. Matrices tt E are also used in cryptography. This mathematical tool is not only used in certain branches of sciences, but also in genetics, economics, sociology, modern psychology and industrial C management. In this chapter, we shall find it interesting to become acquainted with the no N fundamentals of matrix and matrix algebra. 3.2 Matrix Suppose we wish to express the information that Radha has 15 notebooks. We may © express it as with the understanding that the number inside [ ] is the number of notebooks that Radha has. Now, if we have to express that Radha has 15 notebooks and 6 pens. We may express it as [15 6] with the understanding that first number inside [ ] is the number of notebooks while the other one is the number of pens possessed by Radha. Let us now suppose that we wish to express the information of possession  MATRICES 57 of notebooks and pens by Radha and her two friends Fauzia and Simran which is as follows: Radha has 15 notebooks and 6 pens, Fauzia has 10 notebooks and 2 pens, Simran has 13 notebooks and 5 pens. he Now this could be arranged in the tabular form as follows: Notebooks Pens Radha 15 6 Fauzia 10 2 is Simran 13 5 and this can be expressed as bl pu be T re o R or Radha Fauzia Simran tt E Notebooks 15 10 13 Pens 6 2 5 C which can be expressed as: no N © In the first arrangement the entries in the first column represent the number of note books possessed by Radha, Fauzia and Simran, respectively and the entries in the second column represent the number of pens possessed by Radha, Fauzia and Simran,  58 MATHEMATICS respectively. Similarly, in the second arrangement, the entries in the first row represent the number of notebooks possessed by Radha, Fauzia and Simran, respectively. The entries in the second row represent the number of pens possessed by Radha, Fauzia and Simran, respectively. An arrangement or display of the above kind is called a matrix. Formally, we define matrix as: Definition 1 A matrix is an ordered rectangular array of numbers or functions. The he numbers or functions are called the elements or the entries of the matrix. We denote matrices by capital letters. The following are some examples of matrices: ⎡ 1⎤ ⎡– 2 5⎤ ⎢2 + i 3 − 2 ⎥ is ⎢ ⎥ ⎢ ⎥ ⎡1 + x x3 3 ⎤ A=⎢ 0 5 ⎥ , B = ⎢ 3.5 –1 2 ⎥ , C = ⎢ ⎥ ⎢ ⎥ ⎣ cos x sin x + 2 tan x ⎦ ⎢3 ⎥ 5 ⎣ 6⎦ bl ⎢ 3 5 ⎥ ⎣ 7 ⎦ In the above examples, the horizontal lines of elements are said to constitute, rows pu of the matrix and the vertical lines of elements are said to constitute, columns of the matrix. Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2 be T rows and 3 columns. re 3.2.1 Order of a matrix o R A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix). So referring to the above examples of matrices, we have A as 3 × 2 matrix, B as 3 × 3 matrix and C as 2 × 3 matrix. We observe that A has tt E 3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively. In general, an m × n matrix has the following rectangular array: C no N © or A = [aij]m × n, 1≤ i ≤ m, 1≤ j ≤ n i, j ∈ N Thus the ith row consists of the elements ai1, ai2, ai3,..., ain, while the jth column consists of the elements a1j, a2j, a3j,..., amj , In general aij, is an element lying in the ith row and jth column. We can also call it as the (i, j)th element of A. The number of elements in an m × n matrix will be equal to mn.  MATRICES 59 $Note In this chapter 1. We shall follow the notation, namely A = [aij]m × n to indicate that A is a matrix of order m × n. 2. We shall consider only those matrices whose elements are real numbers or functions taking real values. he We can also represent any point (x, y) in a plane by a matrix (column or row) as ⎡x⎤ ⎢ y ⎥ (or [x, y]). For example point P(0, 1) as a matrix representation may be given as ⎣ ⎦ is ⎡0 ⎤ P = ⎢ ⎥ or [0 1]. bl ⎣1 ⎦ Observe that in this way we can also express the vertices of a closed rectilinear pu figure in the form of a matrix. For example, consider a quadrilateral ABCD with vertices A (1, 0), B (3, 2), C (1, 3), D (–1, 2). Now, quadrilateral ABCD in the matrix form, can be represented as be T A⎡ 1 0⎤ re A B C D B ⎢⎢ 3 2 ⎥⎥ o R ⎡1 3 1 −1⎤ X=⎢ ⎥ or Y= ⎣ 0 2 3 2⎦ 2 × 4 C⎢ 1 3⎥ ⎢ ⎥ D ⎣−1 tt E 2 ⎦ 4× 2 Thus, matrices can be used as representation of vertices of geometrical figures in C a plane. Now, let us consider some examples. no N Example 1 Consider the following information regarding the number of men and women workers in three factories I, II and III Men workers Women workers © I 30 25 II 25 31 III 27 26 Represent the above information in the form of a 3 × 2 matrix. What does the entry in the third row and second column represent?  60 MATHEMATICS Solution The information is represented in the form of a 3 × 2 matrix as follows: ⎡ 30 25⎤ A = ⎢⎢ 25 31⎥⎥ ⎢⎣ 27 26 ⎥⎦ The entry in the third row and second column represents the number of women he workers in factory III. Example 2 If a matrix has 8 elements, what are the possible orders it can have? Solution We know that if a matrix is of order m × n, it has mn elements. Thus, to find is all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural numbers, whose product is 8. bl Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4) Hence, possible orders are 1 × 8, 8 ×1, 4 × 2, 2 × 4 1 pu Example 3 Construct a 3 × 2 matrix whose elements are given by aij = 2 |i −3j |. ⎡ a11 a12 ⎤ be T Solution In general a 3 × 2 matrix is given by A = ⎢ a21 a22 ⎥. ⎢ ⎥ re ⎢⎣ a31 a32 ⎥⎦ o R 1 Now aij = | i − 3 j | , i = 1, 2, 3 and j = 1, 2. 2 tt E 1 1 5 Therefore a11 = |1 − 3 × 1| = 1 a12 = |1 − 3 × 2 | = 2 2 2 C 1 1 1 a21 = | 2 − 3 × 1| = a22 = | 2 − 3× 2 | = 2 2 2 2 no N 1 1 3 a31 = | 3 − 3 × 1| = 0 a32 = | 3 − 3× 2 | = 2 2 2 © ⎡1 5⎤ ⎢ 2⎥ ⎢1 ⎥ Hence the required matrix is given by A = ⎢ 2⎥. ⎢2 3⎥ ⎢0 ⎥ ⎣ 2⎦  MATRICES 61 3.3 Types of Matrices In this section, we shall discuss different types of matrices. (i) Column matrix A matrix is said to be a column matrix if it has only one column. ⎡ 0 ⎤ he ⎢ ⎥ ⎢ 3⎥ For example, A = ⎢ −1 ⎥ is a column matrix of order 4 × 1. ⎢ ⎥ ⎣⎢1/ 2 ⎦⎥ is In general, A = [aij] m × 1 is a column matrix of order m × 1. (ii) Row matrix bl A matrix is said to be a row matrix if it has only one row. ⎡ 1 ⎤ pu For example, B = ⎢ − ⎣ 2 5 2 3⎥ is a row matrix. ⎦1× 4 In general, B = [bij] 1 × n is a row matrix of order 1 × n. be T (iii) Square matrix re A matrix in which the number of rows are equal to the number of columns, is o R said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. tt E ⎡ 3 −1 0⎤ ⎢3 ⎥ = ⎢ 1 ⎥ is a square matrix of order 3. C For example A 3 2 ⎢2 ⎥ ⎢ −1⎦⎥ ⎣4 3 no N In general, A = [aij] m × m is a square matrix of order m. $Note If A = [a ] is a square matrix of order n, then elements (entries) a , a ,..., a ij 11 22 nn © ⎡ 1 −3 1 ⎤ ⎢ ⎥ are said to constitute the diagonal, of the matrix A. Thus, if A = ⎢ 2 4 −1⎥. ⎣⎢ 3 5 6 ⎥⎦ Then the elements of the diagonal of A are 1, 4, 6.  62 MATHEMATICS (iv) Diagonal matrix A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal matrix if bij = 0, when i ≠ j. ⎡ −1.1 0 0 ⎤ ⎡ −1 0 ⎤ ⎢ For example, A = , B = ⎢ ⎥, C=⎢ 0 2 0 ⎥⎥ , are diagonal matrices he ⎣ 0 2 ⎦ ⎢⎣ 0 0 3⎥⎦ of order 1, 2, 3, respectively. (v) Scalar matrix is A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [bij] n × n is said to be a scalar matrix if bij = 0, when i ≠ j bl bij = k, when i = j, for some constant k. For example pu ⎡ 3 0 0⎤ ⎡ −1 0 ⎤ ⎢ ⎥ A = , B=⎢ ⎥, C=⎢ 0 3 0⎥ be T ⎣ 0 −1⎦ ⎢ ⎥ ⎣0 0 3⎦ re o R are scalar matrices of order 1, 2 and 3, respectively. (vi) Identity matrix tt E A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. In other words, the square matrix A = [aij] n × n is an ⎧1 if i = j C identity matrix, if aij = ⎨. ⎩0 if i ≠ j no N We denote the identity matrix of order n by In. When order is clear from the context, we simply write it as I. ⎡1 0 0 ⎤ ⎡1 0 ⎤ ⎢0 1 0 ⎥ © For example , ⎢ ⎥, ⎢ ⎥ are identity matrices of order 1, 2 and 3, ⎣ 0 1 ⎦ ⎢0 0 1 ⎥ ⎣ ⎦ respectively. Observe that a scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix.  MATRICES 63 (vii) Zero matrix A matrix is said to be zero matrix or null matrix if all its elements are zero. ⎡0 0 ⎤ ⎡0 0 0 ⎤ For example, , ⎢ ⎥, ⎢ ⎥ , [0, 0] are all zero matrices. We denote ⎣0 0 ⎦ ⎣0 0 0 ⎦ zero matrix by O. Its order will be clear from the context. he 3.3.1 Equality of matrices Definition 2 Two matrices A = [aij] and B = [bij] are said to be equal if (i) they are of the same order is (ii) each element of A is equal to the corresponding element of B, that is aij = bij for all i and j. bl ⎡ 2 3⎤ ⎡ 2 3⎤ ⎡3 2⎤ ⎡ 2 3⎤ For example, ⎢ ⎥ and ⎢ ⎥ are equal matrices but ⎢ ⎥ and ⎢ ⎥ are ⎣ 0 1⎦ ⎣ 0 1⎦ ⎣0 1 ⎦ ⎣ 0 1⎦ not equal matrices. Symbolically, if two matrices A and B are equal, we write A = B. pu ⎡ x y ⎤ ⎡ −1.5 0 ⎤ ⎢ ⎥ If ⎢⎢ z a ⎥⎥ = ⎢ 2 be T 6 ⎥ , then x = – 1.5, y = 0, z = 2, a = 6 , b = 3, c = 2 ⎣⎢b c ⎥⎦ ⎢⎣3 2 ⎥⎦ re o R ⎡ x + 3 z + 4 2 y − 7⎤ ⎡ 0 6 3y − 2 ⎤ ⎢ −6 a −1 ⎥ ⎢ 0 ⎥ = ⎢− 6 −3 2c + 2 ⎥⎥ tt E Example 4 If ⎢ ⎢⎣b − 3 − 21 0 ⎥⎦ ⎢⎣ 2b + 4 − 21 0 ⎥⎦ C Find the values of a, b, c, x, y and z. Solution As the given matrices are equal, therefore, their corresponding elements must be equal. Comparing the corresponding elements, we get no N x + 3 = 0, z + 4 = 6, 2y – 7 = 3y – 2 a – 1 = – 3, 0 = 2c + 2 b – 3 = 2b + 4, Simplifying, we get © a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2 Example 5 Find the values of a, b, c, and d from the following equation: ⎡ 2a + b a − 2b ⎤ ⎡ 4 −3⎤ ⎢5c − d 4c + 3d ⎥ = ⎢11 24 ⎥ ⎣ ⎦ ⎣ ⎦  64 MATHEMATICS Solution By equality of two matrices, equating the corresponding elements, we get 2a + b = 4 5c – d = 11 a – 2b = – 3 4c + 3d = 24 Solving these equations, we get a = 1, b = 2, c = 3 and d = 4 he EXERCISE 3.1 ⎡2 5 19 −7 ⎤ ⎢ ⎥ is 5 1. In the matrix A = ⎢ 35 −2 12 ⎥ , write: ⎢ 2 ⎥ ⎢ ⎥ ⎣ 3 1 −5 17 ⎦ bl (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements a13, a21, a33, a24, a23. pu 2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements? be T 3. If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements? re o R 4. Construct a 2 × 2 matrix, A = [aij], whose elements are given by: (i + j ) 2 i (i + 2 j ) 2 (i) aij = (ii) aij = (iii) aij = tt E 2 j 2 5. Construct a 3 × 4 matrix, whose elements are given by: C 1 (i) aij = | −3i + j | (ii) aij = 2i − j 2 no N 6. Find the values of x, y and z from the following equations: ⎡ x + y + z ⎤ ⎡9 ⎤ ⎡ 4 3⎤ ⎡ y z ⎤ ⎡x + y 2 ⎤ ⎡6 2⎤ ⎢ x + z ⎥ = ⎢5⎥ (i) ⎢ ⎥=⎢ ⎥ (ii) ⎢ = xy ⎥⎦ ⎢⎣ 5 8 ⎥⎦ (iii) ⎢ ⎥ ⎢ ⎥ ⎣5 + z © ⎣ x 5⎦ ⎣ 1 5 ⎦ ⎢⎣ y + z ⎥⎦ ⎢⎣7 ⎥⎦ 7. 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⎥ ⎣ ⎦ ⎣ ⎦  MATRICES 65 8. A = [aij]m × n\ is a square matrix, if (A) m < n (B) m > n (C) m = n (D) None of these 9. Which of the given values of x and y make the following pair of matrices equal ⎡3 x + 7 5 ⎤ ⎡0 y − 2⎤ ⎢ y + 1 2 − 3 x ⎥ , ⎢8 4 ⎥⎦ ⎣ ⎦ ⎣ −1 he (A) x = , y=7 (B) Not possible to find 3 −2 1 2 (C) y = 7, x = (D) x , y 3 3 3 is 10. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is: (A) 27 (B) 18 (C) 81 (D) 512 bl 3.4 Operations on Matrices In this section, we shall introduce certain operations on matrices, namely, addition of matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices. pu 3.4.1 Addition of matrices Suppose Fatima has two factories at places A and B. Each factory produces sport be T shoes for boys and girls in three different price categories labelled 1, 2 and 3. The quantities produced by each factory are represented as matrices given below: re o R tt E C Suppose Fatima wants to know the total production of sport shoes in each price no N category. Then the total production In category 1 : for boys (80 + 90), for girls (60 + 50) In category 2 : for boys (75 + 70), for girls (65 + 55) © In category 3 : for boys (90 + 75), for girls (85 + 75) ⎡80 + 90 60 + 50 ⎤ This can be represented in the matrix form as ⎢ 75 + 70 65 + 55 ⎥⎥. ⎢ ⎢⎣90 + 75 85 + 75 ⎥⎦  66 MATHEMATICS This new matrix is the sum of the above two matrices. We observe that the sum of two matrices is a matrix obtained by adding the corresponding elements of the given matrices. Furthermore, the two matrices have to be of the same order. ⎡ a11 a12 a13 ⎤ ⎡b11 b12 b13 ⎤ Thus, if A = ⎢ ⎥ is a 2 × 3 matrix and B = ⎢ ⎥ is another ⎣ a21 a22 a23 ⎦ ⎣b21 b22 b23 ⎦ he ⎡ a11 + b11 a12 + b12 a13 + b13 ⎤ 2×3 matrix. Then, we define A + B = ⎢ ⎥. ⎣ a21 + b21 a22 + b22 a23 + b23 ⎦ is In general, if A = [aij] and B = [bij] are two matrices of the same order, say m × n. Then, the sum of the two matrices A and B is defined as a matrix C = [cij]m × n, where cij = aij + bij, for all possible values of i and j. bl ⎡ 3 1 − 1⎤ ⎡2 5 1⎤ Example 6 Given A = ⎢ ⎢ ⎥ and B = ⎢ ⎥ 1 ⎥ , find A + B pu ⎣ 2 3 0 ⎦ ⎢⎣ −2 3 2 ⎥⎦ be T Since A, B are of the same order 2 × 3. Therefore, addition of A and B is defined and is given by re o R ⎡ 2 + 3 1 + 5 1 − 1⎤ ⎡ 2 + 3 1 + 5 0 ⎤ A+B = ⎢ ⎥ ⎢ 1⎥ = ⎢ ⎥ ⎢2 − 2 3+3 0+ 0 6 1⎥ tt E ⎢⎣ 2⎥⎦ ⎢⎣ 2 ⎥⎦ $ Note C 1. We emphasise that if A and B are not of the same order, then A + B is not no N ⎡ 2 3⎤ ⎡1 2 3⎤ defined. For example if A = ⎢ ⎥ , B=⎢ ⎥ , then A + B is not defined. ⎣1 0 ⎦ ⎣1 0 1 ⎦ 2. We may observe that addition of matrices is an example of binary operation © on the set of matrices of the same order. 3.4.2 Multiplication of a matrix by a scalar Now suppose that Fatima has doubled the production at a factory A in all categories (refer to 3.4.1).  MATRICES 67 Previously quantities (in standard units) produced by factory A were he Revised quantities produced by factory A are as given below: Boys Girls 1 ⎡ 2 × 80 2 × 60 ⎤ is 2 ⎢⎢ 2 × 75 2 × 65⎥⎥ 3 ⎢⎣ 2 × 90 2 × 85 ⎥⎦ bl ⎡160 120 ⎤ pu This can be represented in the matrix form as ⎢150 ⎢ 130 ⎥⎥. We observe that ⎢⎣180 170 ⎥⎦ be T the new matrix is obtained by multiplying each element of the previous matrix by 2. re In general, we may define multiplication of a matrix by a scalar as follows: if o R A = [aij] m × n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k. tt E In other words, kA = k [aij] m × n = [k (aij)] m × n, that is, (i, j)th element of kA is kaij for all possible values of i and j. C ⎡ 3 1 1.5⎤ ⎢ ⎥ For example, if A = ⎢ 5 7 −3 ⎥ , then no N ⎢2 0 5⎥ ⎣ ⎦ ⎡ 3 1 1.5⎤ ⎡ 9 3 4.5⎤ © ⎢ ⎥ ⎢ ⎥ 3A = 3 ⎢ 5 7 −3 ⎥ = ⎢3 5 21 −9 ⎥ ⎢2 0 5⎥ ⎢ 6 0 15 ⎥⎦ ⎣ ⎦ ⎣ Negative of a matrix The negative of a matrix is denoted by – A. We define – A = (– 1) A.  68 MATHEMATICS ⎡ 3 1⎤ For example, let A= ⎢ ⎥ , then – A is given by ⎣ −5 x ⎦ ⎡ 3 1 ⎤ ⎡ −3 − 1 ⎤ – A = (– 1) A = (−1) ⎢ ⎥=⎢ ⎥ ⎣ −5 x ⎦ ⎣ 5 − x ⎦ he Difference of matrices If A = [aij], B = [bij] are two matrices of the same order, say m × n, then difference A – B is defined as a matrix D = [dij], where dij = aij – bij, for all value of i and j. In other words, D = A – B = A + (–1) B, that is sum of the matrix A and the matrix – B. is ⎡1 2 3⎤ ⎡ 3 −1 3 ⎤ Example 7 If A = ⎢ ⎥ and B = ⎢ ⎥ , then find 2A – B. ⎣ 2 3 1⎦ ⎣ −1 0 2 ⎦ bl Solution We have 1 2 3 3 1 3 2A – B = 2 pu 2 3 1 1 0 2 ⎡ 2 4 6 ⎤ ⎡ −3 1 −3⎤ = ⎢ ⎥+⎢ ⎥ ⎣ 4 6 2 ⎦ ⎣ 1 0 −2 ⎦ be T re ⎡ 2 − 3 4 + 1 6 − 3 ⎤ ⎡ −1 5 3 ⎤ ⎥=⎢ o R = ⎢ ⎥ ⎣ 4 + 1 6 + 0 2 − 2⎦ ⎣ 5 6 0⎦ 3.4.3 Properties of matrix addition tt E The addition of matrices satisfy the following properties: (i) Commutative Law If A = [aij], B = [bij] are matrices of the same order, say C m × n, then A + B = B + A. Now A + B = [aij] + [bij] = [aij + bij] no N = [bij + aij] (addition of numbers is commutative) = ([bij] + [aij]) = B + A (ii) Associative Law For any three matrices A = [aij], B = [bij], C = [cij] of the same order, say m × n, (A + B) + C = A + (B + C). © Now (A + B) + C = ([aij] + [bij]) + [cij] = [aij + bij] + [cij] = [(aij + bij) + cij] = [aij + (bij + cij)] (Why?) = [aij] + [(bij + cij)] = [aij] + ([bij] + [cij]) = A + (B + C)  MATRICES 69 (iii) Existence of additive identity Let A = [a ij] be an m × n matrix and O be an m × n zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition. (iv) The existence of additive inverse Let A = [aij]m × n be any matrix, then we have another matrix as – A = [– aij]m × n such that A + (– A) = (– A) + A= O. So – A is the additive inverse of A or negative of A. he 3.4.4 Properties of scalar multiplication of a matrix If A = [aij] and B = [bij] be two matrices of the same order, say m × n, and k and l are scalars, then is (i) k(A +B) = k A + kB, (ii) (k + l)A = k A + l A (ii) k (A + B) = k ([aij] + [bij]) bl = k [aij + bij] = [k (aij + bij)] = [(k aij) + (k bij)] = [k aij] + [k bij] = k [aij] + k [bij] = kA + kB (iii) ( k + l) A = (k + l) [aij] pu = [(k + l) aij] + [k aij] + [l aij] = k [aij] + l [aij] = k A + l A be T ⎡8 0 ⎤ ⎡ 2 −2 ⎤ re ⎢ ⎥ ⎢ ⎥ Example 8 If A = ⎢ 4 −2 ⎥ and B = ⎢ 4 2 ⎥ , then find the matrix X, such that o R ⎢⎣ 3 6 ⎥⎦ ⎢⎣ −5 1 ⎥⎦ 2A + 3X = 5B. tt E Solution We have 2A + 3X = 5B C or 2A + 3X – 2A = 5B – 2A or 2A – 2A + 3X = 5B – 2A (Matrix addition is commutative) no N or O + 3X = 5B – 2A (– 2A is the additive inverse of 2A) or 3X = 5B – 2A (O is the additive identity) 1 or X= (5B – 2A) © 3 ⎛ ⎡ 2 −2 ⎤ ⎡8 0 ⎤ ⎞ ⎛ ⎡ 10 −10 ⎤ ⎡ −16 0 ⎤ ⎞ 1⎜ ⎢ ⎥ ⎢ ⎥ ⎟ 1 ⎜⎢ ⎥ ⎢ −8 4 ⎥ ⎟ or X = ⎜ 5 ⎢ 4 2 ⎥ − 2 ⎢ 4 −2 ⎥ ⎟ = ⎜ ⎢ 20 10 ⎥ + ⎢ ⎥⎟ 3⎜ 3 ⎜ ⎢ −25 5 ⎥ ⎝ ⎢⎣ −5 1 ⎥⎦ ⎢⎣ 3 6 ⎥⎦ ⎟⎠ ⎝⎣ ⎦ ⎢⎣ −6 −12 ⎥⎦ ⎟⎠  70 MATHEMATICS ⎡ −10 ⎤ ⎢ −2 3 ⎥ ⎡ 10 − 16 −10 + 0 ⎤ ⎡ − 6 −10 ⎤ ⎢ ⎥ 1⎢ ⎥ 1⎢ ⎥ ⎢ 4 14 ⎥ = ⎢ 20 − 8 10 + 4 ⎥ = 3⎢ 12 14 ⎥ = ⎢ 3 3 ⎥ ⎢⎣ −25 − 6 5 − 12 ⎥⎦ ⎢⎣ −31 −7 ⎥⎦ ⎢ ⎥ ⎢ −31 −7 ⎥ he ⎢⎣ 3 3 ⎥⎦ ⎡5 2⎤ ⎡3 6 ⎤ Example 9 Find X and Y, if X + Y = ⎢ ⎥ and X − Y = ⎢ ⎥. ⎣0 9 ⎦ ⎣ 0 −1⎦ is ⎡5 2⎤ ⎡3 6 ⎤ Solution We have ( X + Y ) + ( X − Y ) = ⎢ ⎥+⎢ ⎥. ⎣ 0 9 ⎦ ⎣0 −1⎦ bl ⎡8 8⎤ ⎡8 8⎤ or (X + X) + (Y – Y) = ⎢ ⎥ ⇒ 2X = ⎢ ⎥ pu ⎣ 0 8⎦ ⎣0 8⎦ 1 ⎡8 8⎤ ⎡ 4 4 ⎤ or X= ⎢0 8⎥ = ⎢ 0 4 ⎥ be T 2 ⎣ ⎦ ⎣ ⎦ re ⎡5 2⎤ ⎡3 6 ⎤ o R Also (X + Y) – (X – Y) = ⎢ ⎥−⎢ ⎥ ⎣ 0 9 ⎦ ⎣0 −1⎦ tt E ⎡5 − 3 2 − 6 ⎤ ⎡ 2 −4 ⎤ or (X – X) + (Y + Y) = ⎢ ⎥ ⇒ 2Y = ⎢ ⎥ ⎣ 0 9 + 1⎦ ⎣ 0 10 ⎦ C 1 ⎡ 2 − 4 ⎤ ⎡ 1 −2 ⎤ or Y= ⎢ 0 10 ⎥ = ⎢ 0 5 ⎥ 2 ⎣ ⎦ ⎣ ⎦ no N Example 10 Find the values of x and y from the following equation: ⎡x 5 ⎤ ⎡3 −4 ⎤ ⎡7 6⎤ 2⎢ ⎥ +⎢ ⎥ = ⎢ ⎥ ⎣7 y − 3 ⎦ ⎣1 2 ⎦ ⎣15 14 ⎦ © Solution We have ⎡x 5 ⎤ ⎡3 −4 ⎤ ⎡7 6⎤ ⎡2 x 10 ⎤ ⎡3 − 4 ⎤ ⎡ 7 6 ⎤ + ⇒ ⎢ ⎥+⎢ = 2⎢ y − 3⎥⎦ ⎢⎣1 2 ⎥⎦ = ⎢ ⎥ ⎣7 ⎣15 14 ⎦ ⎣14 2 y − 6 ⎦ ⎣1 2 ⎥⎦ ⎢⎣15 14 ⎥⎦  MATRICES 71 ⎡2x + 3 10 − 4 ⎤ ⎡7 6⎤ ⎡2 x + 3 6 ⎤ ⎡7 6⎤ ⎢ 14 + 1 2 y − 6 + 2 ⎥ = ⎢ ⇒ ⎢ = ⎥ 2 y − 4 ⎥⎦ ⎢⎣15 14 ⎥⎦ or ⎣ ⎦ ⎣15 14 ⎦ ⎣ 15 or 2x + 3 = 7 and 2y – 4 = 14 (Why?) or 2x = 7 – 3 and 2y = 18 4 18 he or x= and y= 2 2 i.e. x =2 and y = 9. Example 11 Two farmers Ramkishan and Gurcharan Singh cultivates only three is varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B. bl pu be T re o R tt E (i) Find the combined sales in September and October for each farmer in each C variety. (ii) Find the decrease in sales from September to October. no N (iii) If both farmers receive 2% profit on gross sales, compute the profit for each farmer and for each variety sold in October. Solution (i) Combined sales in September and October for each farmer in each variety is © given by  72 MATHEMATICS (ii) Change in sales from September to October is given by he 2 (iii) 2% of B = × B = 0.02 × B 100 is = 0.02 bl = pu Thus, in October Ramkishan receives Rs 100, Rs 200 and Rs 120 as profit in the sale of each variety of rice, respectively, and Grucharan Singh receives profit of Rs be T 400, Rs 200 and Rs 200 in the sale of each variety of rice, respectively. re o R 3.4.5 Multiplication of matrices Suppose Meera and Nadeem are two friends. Meera wants to buy 2 pens and 5 story books, while Nadeem needs 8 pens and 10 story books. They both go to a shop to tt E enquire about the rates which are quoted as follows: Pen – Rs 5 each, story book – Rs 50 each. C How much money does each need to spend? Clearly, Meera needs Rs (5 × 2 + 50 × 5) that is Rs 260, while Nadeem needs (8 × 5 + 50 × 10) Rs, that is Rs 540. In terms of matrix representation, we can write the above information as follows: no N Requirements Prices per piece (in Rupees) Money needed (in Rupees) ⎡2 5 ⎤ ⎡5⎤ ⎡ 5 × 2 + 5 × 50 ⎤ ⎡ 260 ⎤ ⎢ 8 10 ⎥ ⎢50 ⎥ ⎢ 8 × 5 + 10 × 50⎥ = ⎢ 540 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ © Suppose that they enquire about the rates from another shop, quoted as follows: pen – Rs 4 each, story book – Rs 40 each. Now, the money required by Meera and Nadeem to make purchases will be respectively Rs (4 × 2 + 40 × 5) = Rs 208 and Rs (8 × 4 + 10 × 40) = Rs 432  MATRICES 73 Again, the above information can be represented as follows: Requirements Prices per piece (in Rupees) Money needed (in Rupees) ⎡2 5 ⎤ ⎡4⎤ ⎡ 4 × 2 + 40 × 5 ⎤ ⎡ 208 ⎤ ⎢ 8 10 ⎥ ⎢ 40 ⎥ ⎢ 8 × 4 + 10 × 4 0⎥ = ⎢ 432 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Now, the information in both the cases can be combined and expressed in terms of he matrices as follows: Requirements Prices per piece (in Rupees) Money needed (in Rupees) ⎡2 5 ⎤ ⎡5 4⎤ ⎡ 5 × 2 + 5 × 50 4 × 2 + 40 × 5 ⎤ ⎢50 40 ⎥ ⎢ 8 × 5 + 10 × 5 0 8 × 4 + 10 × 4 0⎥ is ⎢ 8 10 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ 260 208 ⎤ bl = ⎢ ⎥ ⎣ 540 432 ⎦ The above is an example of multiplication of matrices. We observe that, for pu multiplication of two matrices A and B, the number of columns in A should be equal to the number of rows in B. Furthermore for getting the elements of the product matrix, be T we take rows of A and columns of B, multiply them element-wise and take the sum. Formally, we define multiplication of matrices as follows: re The product of two matrices A and B is defined if the number of columns of A is o R equal to the number of rows of B. Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. tt E To get the (i, k)th element cik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products. In other words, if A = [aij]m × n, B = [bjk]n × p, then the ith row of A is [ai1 ai2... ain] and the kth column of C ⎡ b1k ⎤ ⎢b ⎥ n ⎢ 2k ⎥ no N B is ⎢.. ⎥ , then cik = ai1 b1k + ai2 b2k + ai3 b3k +... + ain bnk = ∑ aij b jk. j =1 ⎢. ⎥ ⎢b ⎥ ⎣ nk ⎦ © The matrix C = [cik]m × p is the product of A and B. ⎡ 2 7⎤ ⎡1 −1 2 ⎤ ⎢ −1 1 ⎥ , then the product CD is defined For example, if C = ⎢ ⎥ and D = ⎢ ⎥ ⎣0 3 4 ⎦ ⎢⎣ 5 − 4 ⎥⎦  74 MATHEMATICS ⎡ 2 7⎤ ⎡1 −1 2⎤ ⎢ ⎥ and is given by CD = ⎢ ⎥ ⎢ −1 1 ⎥. This is a 2 × 2 matrix in which each ⎣ 0 3 4 ⎦ ⎢ 5 − 4⎥ ⎣ ⎦ entry is the sum of the products across some row of C with the corresponding entries he down some column of D. These four computations are is bl pu be T re o R ⎡13 −2 ⎤ tt E Thus CD = ⎢ ⎥ ⎣17 −13⎦ C ⎡6 9⎤ ⎡ 2 6 0⎤ Example 12 Find AB, if A = ⎢ ⎥ and B = ⎢ ⎥. ⎣ 2 3⎦ ⎣7 9 8 ⎦ no N Solution The matrix A has 2 columns which is equal to the number of rows of B. Hence AB is defined. Now ⎡ 6(2) + 9(7) 6(6) + 9(9) 6(0) + 9(8) ⎤ © AB = ⎢ ⎥ ⎣ 2(2) + 3(7) 2(6) + 3(9) 2(0) + 3(8) ⎦ ⎡12 + 63 36 + 81 0 + 72 ⎤ ⎡ 75 117 72 ⎤ =⎢ ⎥ = ⎢ ⎥ ⎣ 4 + 21 12 + 27 0 + 24⎦ ⎣ 25 39 24 ⎦  MATRICES 75 Remark If AB is defined, then BA need not be defined. In the above example, AB is defined but BA is not defined because B has 3 column while A has only 2 (and not 3) rows. If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined if and only if n = k and l = m. In particular, if both A and B are square matrices of the same order, then both AB and BA are defined. Non-commutativity of multiplication of matrices he Now, we shall see by an example that even if AB and BA are both defined, it is not necessary that AB = BA. ⎡ 2 3⎤ ⎡ 1 −2 3⎤ and B = ⎢ 4 5⎥⎥ , then find AB, BA. Show that ⎢ is Example 13 If A = ⎢ ⎥ ⎣− 4 2 5 ⎦ ⎢⎣ 2 1⎥⎦ bl AB ≠ BA. Solution Since A is a 2 × 3 matrix and B is 3 × 2 matrix. Hence AB and BA are both defined and are matrices of order 2 × 2 and 3 × 3, respectively. Note that pu ⎡ 2 3⎤ ⎡ 1 −2 3⎤ ⎢ ⎥ ⎡ 2−8+ 6 3 − 10 + 3 ⎤ ⎡ 0 − 4 ⎤ AB = ⎢ ⎢ ⎥=⎢ 3 ⎥⎦ be T ⎥ 5⎦ ⎢ 4 5 ⎥ = ⎣− 4 2 ⎣ −8 + 8 + 10 −12 + 10 + 5⎦ ⎣10 ⎢⎣ 2 1⎥⎦ re o R ⎡2 3⎤ ⎡ 2 − 12 − 4 + 6 6 + 15 ⎤ ⎡ −10 2 21⎤ ⎡ 1 −2 3 ⎤ ⎢ ⎢ ⎥ and BA = ⎢⎢ 4 ⎥ 5⎥ ⎢ ⎥ = ⎢ 4 − 20 −8 + 10 12 + 25⎥⎥ = ⎢ −16 2 37 ⎥ −4 2 5 ⎦ tt E ⎢⎣ 2 1⎥⎦ ⎣ ⎢⎣ 2 − 4 − 4 + 2 6 + 5 ⎥⎦ ⎢⎣ −2 −2 11 ⎥⎦ Clearly AB ≠ BA C In the above example both AB and BA are of different order and so AB ≠ BA. But one may think that perhaps AB and BA could be the same if they were of the same no N order. But it is not so, here we give an example to show that even if AB and BA are of same order they may not be same. ⎡1 0 ⎤ ⎡0 1 ⎤ ⎡ 0 1⎤ Example 14 If A = ⎢ ⎥ and B = ⎢ ⎥ , then AB = ⎢ ⎥. © ⎣ 0 −1⎦ ⎣1 0⎦ ⎣ −1 0 ⎦ ⎡ 0 −1⎤ and BA = ⎢ ⎥. Clearly AB ≠ BA. ⎣1 0 ⎦ Thus matrix multiplication is not commutative.  76 MATHEMATICS $ Note This does not mean that AB ≠ BA for every pair of matrices A, B for which AB and BA, are defined. For instance, ⎡1 0 ⎤ ⎡3 0⎤ ⎡3 0⎤ If A = ⎢ ⎥ , B=⎢ ⎥ , then AB = BA = ⎢ ⎥ ⎣0 2⎦ ⎣0 4⎦ ⎣0 8 ⎦ Observe that multiplication of diagonal matrices of same order will be commutative. he Zero matrix as the product of two non zero matrices We know that, for real numbers a, b if ab = 0, then either a = 0 or b = 0. This need not be true for matrices, we will observe this through an example. is ⎡ 0 −1⎤ ⎡3 5⎤ Example 15 Find AB, if A = ⎢ ⎥ and B = ⎢ ⎥. ⎣0 2⎦ ⎣0 0⎦ bl ⎡ 0 −1 ⎤ ⎡ 3 5 ⎤ ⎡ 0 0 ⎤ Solution We have AB = ⎢ ⎥⎢ ⎥=⎢ ⎥. ⎣0 2 ⎦ ⎣0 0⎦ ⎣0 0 ⎦ pu Thus, if the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix. be T 3.4.6 Properties of multiplication of matrices re The multiplication of matrices possesses the following properties, which we state without o R proof. 1. The associative law For any three matrices A, B and C. We have tt E (AB) C = A (BC), whenever both sides of the equality are defined. 2. The distributive law For three matrices A, B and C. C (i) A (B+C) = AB + AC (ii) (A+B) C = AC + BC, whenever both sides of equality are defined. no N 3. The existence of multiplicative identity For every square matrix A, there exist an identity matrix of same order such that IA = AI = A. Now, we shall verify these properties by examples. © ⎡1 1 −1⎤ ⎡ 1 3⎤ ⎢ ⎡1 2 3 − 4 ⎤ Example 16 If A = ⎢ 2 0 3 ⎥ , B = ⎢⎢ 0 2⎥⎥ and C = ⎢ ⎥ ⎥ , find ⎢⎣ 3 −1 2 ⎥⎦ ⎢⎣−1 4⎥⎦ ⎣ 2 0 −2 1 ⎦ A(BC), (AB)C and show that (AB)C = A(BC).  MATRICES 77 ⎡1 1 −1⎤ ⎡ 1 3 ⎤ ⎡ 1 + 0 + 1 3 + 2 − 4 ⎤ ⎡ 2 1 ⎤ Solution We have AB = ⎢ 2 0 3 ⎥⎥ ⎢⎢ 0 2 ⎥⎥ = ⎢⎢ 2 + 0 − 3 6 + 0 + 12 ⎥⎥ = ⎢⎢−1 18⎥⎥ ⎢ ⎣⎢ 3 −1 2 ⎥⎦ ⎢⎣−1 4 ⎦⎥ ⎣⎢3 + 0 − 2 9 − 2 + 8 ⎦⎥ ⎣⎢ 1 15⎦⎥ ⎡2 1⎤ ⎡ 2+2 4 + 0 6 − 2 − 8 +1 ⎤ ⎡1 2 3 − 4⎤ ⎢ he ⎢ ⎥ (AB) (C) = ⎢−1 18⎥ ⎢ ⎥ = ⎢ −1 + 36 −2 + 0 −3 − 36 4 + 18⎥⎥ − ⎢⎣ 1 15⎥⎦ ⎣ ⎦ ⎢ + 2 0 2 1 ⎣ 1 30 2 + 0 3 − 30 − 4 + 15⎦⎥ ⎡4 4 4 −7 ⎤ is ⎢35 −2 −39 22 ⎥ = ⎢ ⎥ ⎢⎣31 2 −27 11⎥⎦ bl ⎡ 1 3⎤ ⎡ 1 + 6 2 + 0 3 − 6 −4 + 3 ⎤ ⎡1 2 3 −4 ⎤ ⎢ BC = ⎢ 0 ⎥ ⎥ Now pu ⎢ 2⎥ ⎢ − ⎥ = ⎢ 0 + 4 0 + 0 0 − 4 0 + 2⎥ 4 ⎦⎥ ⎣ ⎦ ⎢− + 2 0 2 1 ⎣⎢ −1 ⎣ 1 8 −2 + 0 −3 − 8 4 + 4 ⎥⎦ be T ⎡7 2 −3 −1 ⎤ re ⎢ ⎥ o R = ⎢ 4 0 −4 2 ⎥ ⎢⎣7 −2 −11 8 ⎥⎦ tt E ⎡1 1 −1 ⎤ ⎡7 2 −3 −1 ⎤ A(BC) = ⎢⎢ 2 0 3 ⎥⎥ ⎢⎢ 4 0 −4 2 ⎥⎥ C Therefore ⎢⎣ 3 −1 2 ⎥⎦ ⎢⎣7 −2 −11 8 ⎥⎦ no N ⎡ 7 + 4 − 7 2 + 0 + 2 −3 − 4 + 11 −1 + 2 − 8 ⎤ ⎢ ⎥ = ⎢14 + 0 + 21 4 + 0 − 6 −6 + 0 − 33 −2 + 0 + 24 ⎥ ⎢⎣ 21 − 4 + 14 6 + 0 − 4 −9 + 4 − 22 −3 − 2 + 16 ⎥⎦ © ⎡4 4 4 −7 ⎤ ⎢35 −2 −39 22 ⎥ = ⎢ ⎥. Clearly, (AB) C = A (BC) ⎢⎣31 2 −27 11⎥⎦  78 MATHEMATICS ⎡ 0 6 7⎤ ⎡0 1 1 ⎤ ⎡2⎤ Example 17 If A = ⎢⎢ − 6 0 8 ⎥ , B = ⎢1 0 2 ⎥ , C = ⎢⎢−2 ⎥⎥ ⎥ ⎢ ⎥ ⎢⎣ 7 −8 0 ⎥⎦ ⎢⎣1 2 0 ⎥⎦ ⎢⎣ 3 ⎥⎦ Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC ⎡ 0 7 8⎤ he ⎢ ⎥ Solution Now, A + B = ⎢−5 0 10 ⎥ ⎣⎢ 8 − 6 0 ⎥⎦ is ⎡ 0 7 8⎤ ⎡2 ⎤ ⎡ 0 − 14 + 24 ⎤ ⎡10 ⎤ ⎢ 10 ⎥⎥ ⎢−2 ⎥ = ⎢−10 + 0 + 30 ⎥ = ⎢ 20 ⎥ So (A + B) C = ⎢−5 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ 8 − 6 0 ⎥⎦ ⎢⎣ 3 ⎥⎦ ⎢⎣ 16 + 12 + 0 ⎥⎦ ⎢⎣ 28⎥⎦ bl ⎡ 0 6 7 ⎤ ⎡ 2 ⎤ ⎡ 0 − 12 + 21 ⎤ ⎡9⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢12 ⎥ Further pu AC = ⎢ −6 0 8 ⎥ ⎢−2 ⎥ = ⎢−12 + 0 + 24 ⎥ ⎢ ⎥ ⎢⎣ 7 − 8 0 ⎥⎦ ⎢⎣ 3 ⎥⎦ ⎢⎣ 14 + 16 + 0 ⎥⎦ ⎢⎣30 ⎥⎦ be T ⎡0 1 1 ⎤ ⎡ 2 ⎤ ⎡ 0 − 2 + 3⎤ ⎡ 1 ⎤ re ⎢ ⎥ ⎢ ⎥ = ⎢ 2 + 0 + 6⎥ = ⎢ 8 ⎥ BC = ⎢1 0 2 ⎥ ⎢−2 o R and ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣1 2 0 ⎥⎦ ⎢⎣ 3 ⎥⎦ ⎢⎣ 2 − 4 + 0 ⎥⎦ ⎢⎣− 2 ⎥⎦ tt E ⎡9 ⎤ ⎡ 1 ⎤ ⎡10 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ So AC + BC = ⎢12 ⎥ + ⎢ 8 ⎥ = ⎢ 20 ⎥ C ⎢⎣30 ⎥⎦ ⎢⎣−2 ⎥⎦ ⎢⎣ 28⎥⎦ Clearly, (A + B) C = AC + BC no N ⎡1 2 3⎤ ⎢ 1⎥⎥ , then show that A3 – 23A – 40 I = O Example 18 If A = ⎢ 3 −2 ⎢⎣ 4 2 1⎥⎦ © ⎡1 2 3⎤ ⎡ 1 2 3⎤ ⎡19 4 8 ⎤ Solution We have A = A.A = ⎢ 3 −2 2 1⎥⎥ ⎢⎢ 3 −2 1⎥⎥ = ⎢⎢1 12 8 ⎥⎥ ⎢ ⎢⎣ 4 2 1⎥⎦ ⎢⎣ 4 2 1⎥⎦ ⎢⎣14 6 15 ⎥⎦  MATRICES 79 ⎡1 2 3⎤ ⎡19 4 8 ⎤ ⎡ 63 46 69 ⎤ So A = A A = ⎢⎢ 3 −2 3 2 1⎥⎥ ⎢⎢1 12 8 ⎥⎥ = ⎢⎢ 69 −6 23⎥⎥ ⎢⎣ 4 2 1⎥⎦ ⎢⎣14 6 15 ⎥⎦ ⎢⎣92 46 63⎥⎦ Now ⎡ 63 46 69 ⎤ ⎡1 2 3⎤ ⎡1 0 0⎤ he 3 ⎢ 69 −6 23⎥ – 23 ⎢ 3 −2 1⎥ – 40 ⎢⎢ 0 1 0⎥⎥ ⎥ A – 23A – 40I = ⎢ ⎥ ⎢ ⎢⎣92 46 63⎥⎦ ⎢⎣ 4 2 1⎥⎦ ⎢⎣ 0 0 1 ⎥⎦ is ⎡ 63 46 69 ⎤ ⎡ −23 −46 −69 ⎤ ⎡−40 0 0 ⎤ ⎢69 −6 23⎥ + ⎢ −69 46 −23⎥ + ⎢ 0 −40 0 ⎥⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎣92 46 63⎥⎦ ⎢⎣ −92 −46 −23⎥⎦ ⎢⎣ 0 0 −40 ⎥⎦ bl ⎡63 − 23 − 40 46 − 46 + 0 69 − 69 + 0 ⎤ ⎢ ⎥ pu = ⎢69 − 69 + 0 −6 + 46 − 40 23 − 23 + 0 ⎥ ⎢⎣ 92 − 92 + 0 46 − 46 + 0 63 − 23 − 40 ⎥⎦ be T ⎡0 0 0⎤ re ⎢ ⎥ = ⎢0 0 0⎥ = O o R ⎢⎣0 0 0 ⎥⎦ tt E Example 19 In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise) is given in matrix A as C Cost per contact ⎡ 40 ⎤ Telephone ⎢ ⎥ Housecall no N A= ⎢ 100 ⎥ ⎣⎢ 50 ⎦⎥ Letter The number of contacts of each type made in two cities X and Y is given by © Telephone Housecall Letter ⎡1000 500 5000 ⎤ → X B=⎢. Find the total amount spent by the group in the two ⎣3000 1000 10,000⎥⎦ → Y cities X and Y.  80 MATHEMATICS Solution We have ⎡ 40,000 + 50,000 + 250,000 ⎤ → X BA = ⎢ ⎥ ⎣120,000 + 100,000 +500,000 ⎦ → Y ⎡340,000 ⎤ → X = ⎢ ⎥ ⎣ 720,000 ⎦ → Y he So the total amount spent by the group in the two cities is 340,000 paise and 720,000 paise, i.e., Rs 3400 and Rs 7200, respectively. EXERCISE 3.2 is ⎡2 4⎤ ⎡ 1 3⎤ ⎡−2 5⎤ 1. Let A = ⎢ ⎥ ,B=⎢ ⎥ ,C=⎢ ⎣3 2⎦ ⎣−2 5⎦ ⎣3 4 ⎥⎦ bl Find each of the following: (i) A + B (ii) A – B (iii) 3A – C pu (iv) AB (v) BA 2. Compute the following: ⎡a b ⎤ ⎡a b ⎤ ⎡a 2 + b2 b 2 + c 2 ⎤ ⎡ 2ab 2bc ⎤ be T (i) ⎢ + (ii) ⎢ 2 ⎥ + ⎢ ⎥ ⎣−b a ⎥⎦ ⎢⎣ b a ⎥⎦ ⎢⎣ a + c 2 a 2 + b 2 ⎥⎦ ⎣−2ac −2ab ⎦ re o R ⎡−1 4 −6⎤ ⎡12 7 6 ⎤ ⎢ ⎡cos 2 x sin 2 x ⎤ ⎡ sin 2 x cos 2 x ⎤ 5 16 ⎥⎥ + ⎢⎢ 8 0 5 ⎥⎥ (iv) ⎢ (iii) ⎢ 8 ⎥+⎢ 2 ⎥ ⎢⎣ sin x cos x ⎥⎦ ⎢⎣cos x sin x ⎥⎦ 2 2 2 ⎢⎣ 2 8 5 ⎥⎦ ⎢⎣ 3 2 4 ⎥⎦ tt E 3. Compute the indicated products. C ⎡1 ⎤ ⎡a b ⎤ ⎡ a −b ⎤ ⎢ ⎥ ⎡ 1 −2 ⎤ ⎡1 2 3⎤ (i) ⎢ (ii) ⎢ 2⎥ [2 3 4] a ⎥⎦ ⎢⎣ b a ⎥⎦ (iii) ⎢ ⎥⎢ ⎥ ⎣−b ⎢⎣ 3⎥⎦ ⎣ 2 3 ⎦ ⎣ 2 3 1⎦ no N ⎡ 2 3 4 ⎤ ⎡1 −3 5⎤ ⎡2 1⎤ ⎢ ⎡ 1 0 1⎤ (iv) ⎢ 3 4 5 ⎥ ⎢ 0 2 4⎥⎥ 2 ⎥⎥ ⎢ −1 2 1⎥⎦ (v) ⎢ 3 ⎢ ⎥ ⎢ ⎣ © ⎣⎢ 4 5 6 ⎦⎥ ⎢⎣ 3 0 5 ⎥⎦ ⎢⎣−1 1⎦ ⎥ ⎡ 2 −3⎤ ⎡ 3 −1 3 ⎤ ⎢ ⎥ (vi) ⎢ ⎥ ⎢1 0⎥ ⎣−1 0 2 ⎦ ⎢3 1⎥ ⎣ ⎦  MATRICES 81 ⎡1 2 −3⎤ ⎡ 3 −1 2⎤ ⎡4 1 2⎤ ⎢ ⎥ ⎢ 5 ⎥ and C = ⎢⎢ 0 3 ⎥ 2⎥⎥ , then compute 4. If A = ⎢5 0 2 ⎥ , B = ⎢ 4 2 ⎢⎣1 −1 1 ⎥⎦ ⎢⎣ 2 0 3⎥⎦ ⎢⎣ 1 −2 3⎥⎦ (A+B) and (B – C). Also, verify that A + (B – C) = (A + B) – C. he ⎡2 5⎤ ⎡2 3 ⎤ ⎢3 1 1⎥ 3⎥ ⎢5 5 ⎢ ⎥ ⎢ ⎥ 5. If A = ⎢⎢ 4⎥ and B = ⎢ 1 2 1 2 4⎥ ⎥ ⎢ , then compute 3A – 5B. 3 3 3 5 5 5⎥ ⎢ ⎥ ⎢ ⎥ is ⎢ 7 2⎥ ⎢ 7 6 2⎥ 2 ⎣⎢ 3 3 ⎦⎥ ⎢⎣ 5 5 5 ⎦⎥ bl ⎡ cos θ sin θ ⎤ ⎡ sin θ − cos θ ⎤ 6. Simplify cosθ ⎢ ⎥ + sinθ ⎢ ⎣ − sin θ cos θ ⎦ ⎣ cos θ sin θ⎥⎦ pu 7. Find X and Y, if ⎡7 0 ⎤ ⎡3 0⎤ (i) X + Y = ⎢ and X – Y = ⎢ be T ⎥ ⎥ ⎣ 2 5⎦ ⎣0 3⎦ re o R ⎡ 2 3⎤ ⎡ 2 −2 ⎤ (ii) 2X + 3Y = ⎢ ⎥ and 3X + 2Y = ⎢ ⎥ ⎣ 4 0⎦ ⎣ −1 5 ⎦ tt E ⎡3 2 ⎤ ⎡ 1 0⎤ 8. Find X, if Y = ⎢ ⎥ and 2X + Y = ⎢ −3 2 ⎥ ⎣1 4 ⎦ ⎣ ⎦ C ⎡ 1 3 ⎤ ⎡ y 0 ⎤ ⎡5 6 ⎤ 9. Find x and y, if 2 ⎢ ⎥+⎢ ⎥=⎢ ⎥ ⎣ 0 x ⎦ ⎣ 1 2 ⎦ ⎣1 8 ⎦ no N ⎡x z⎤ ⎡ 1 −1 ⎤ ⎡ 3 5⎤ 10. Solve the equation for x, y, z and t, if 2 ⎢ ⎥ +3⎢ ⎥ =3⎢ ⎥ ⎣y t⎦ ⎣0 2 ⎦ ⎣ 4 6⎦ © ⎡ 2⎤ ⎡−1 ⎤ ⎡10 ⎤ 11. If x ⎢ ⎥ + y ⎢ ⎥ = ⎢ ⎥ , find the values of x and y. ⎣ 3⎦ ⎣ 1 ⎦ ⎣5 ⎦ ⎡x y ⎤ ⎡ x 6 ⎤ ⎡ 4 x + y⎤ 12. Given 3 ⎢ ⎥ =⎢ ⎥ +⎢ 3 ⎥⎦ , find the values of x, y, z and w. ⎣ z w ⎦ ⎣ −1 2 w ⎦ ⎣ z + w  82 MATHEMATICS ⎡cos x − sin x 0 ⎤ 13. If F ( x) = ⎢ sin x cos x 0 ⎥ , show that F(x) F(y) = F(x + y). ⎢ ⎥ ⎢⎣ 0 0 1 ⎥⎦ 14. Show that ⎡ 5 −1⎤ ⎡ 2 1 ⎤ ⎡ 2 1 ⎤ ⎡ 5 −1⎤ he (i) ⎢ ⎥⎢ ⎥≠⎢ ⎥⎢ ⎥ ⎣6 7 ⎦ ⎣ 3 4⎦ ⎣ 3 4⎦ ⎣6 7 ⎦ ⎡1 2 3⎤ ⎡−1 1 0 ⎤ ⎡ −1 1 0 ⎤ ⎡1 2 3⎤ ⎢ ⎥⎢ 1 ⎥⎥ ≠ ⎢⎢ 0 −1 1 ⎥⎥ ⎢⎢ 0 1 0⎥⎥ (ii) ⎢0 1 0 ⎥ ⎢ 0 −1 is ⎢⎣1 1 0 ⎥⎦ ⎢⎣ 2 3 4⎥⎦ ⎢⎣ 2 3 4 ⎥⎦ ⎢⎣1 1 0⎥⎦ bl ⎡2 0 1⎤ 15. Find A – 5A + 6I, if A = ⎢⎢ 2 1 2 3⎥⎥ ⎢⎣1 −1 0 ⎥⎦ pu ⎡1 0 2⎤ be T 16. If A = ⎢ 0 2 1 ⎥ , prove that A3 – 6A2 + 7A + 2I = 0 ⎢ ⎥ re ⎢⎣ 2 0 3 ⎥⎦ o R ⎡ 3 −2 ⎤ ⎡1 0 ⎤ 17. If A = ⎢ ⎥ and I= ⎢ ⎥ , find k so that A = kA – 2I 2 ⎣ 4 −2 ⎦ tt E ⎣0 1 ⎦ ⎡ α⎤ − tan ⎥ C ⎢ 0 2 18. If A = ⎢ ⎥ and I is the identity matrix of order 2, show that ⎢ tan α 0 ⎥ ⎢⎣ ⎥⎦ no N 2 ⎡cos α − sin α ⎤ I + A = (I – A) ⎢ ⎣ sin α cos α ⎥⎦ © 19. A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (a) Rs 1800 (b) Rs 2000  MATRICES 83 20. The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra. Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively. Choose the correct answer in Exercises 21 and 22. he 21. The restriction on n, k and p so that PY + WY will be defined are: (A) k = 3, p = n (B) k is arbitrary, p = 2 (C) p is arbitrary, k = 3 (D) k = 2, p = 3 22. If n = p, then the order of the matrix 7X – 5Z is: is (A) p × 2 (B) 2 × n (C) n × 3 (D) p × n 3.5. Transpose of a Matrix bl In this section, we shall learn about transpose of a matrix and special types of matrices such as symmetric and skew symmetric matrices. pu Definition 3 If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A′ or (AT). In other words, if A = [aij]m × n, then A′ = [aji]n × m. For example, be T re ⎡ 3 5⎤ ⎡3 3 0⎤ o R ⎢ ⎥ ⎢ ⎥ if A = ⎢ 3 1 ⎥ , then A′ = ⎢5 1 −1 ⎥ ⎢ 0 −1 ⎥ ⎢⎣ 5 ⎥⎦ 2 × 3 ⎢ ⎥ tt E ⎣ 5 ⎦3 × 2 3.5.1 Properties of transpose of the matrices C We now state the following properties of transpose of matrices without proof. These may be verified by taking suitable examples. no N For any matrices A and B of suitable orders, we have (i) (A′)′ = A, (ii) (kA)′ = kA′ (where k is any constant) (iii) (A + B)′ = A′ + B′ (iv) (A B)′ = B′ A′ © ⎡3 3 2⎤ ⎡ 2 −1 2 ⎤ Example 20 If A = ⎢ ⎥ and B = ⎢ 2 4 ⎥⎦ , verify that ⎣4 2 0⎦ ⎣1 (i) (A′)′ = A, (ii) (A + B)′ = A′ + B′, (iii) (kB)′ = kB′, where k is any constant.  84 MATHEMATICS Solution (i) We have ⎡ 3 4⎤ ⎡3 3 2⎤ ⎢ ⎥ ′ ⎡3 3 2⎤ A= ⎢ ⎥ ⇒ A′ = ⎢ 3 2 ⎥ ⇒ ( A′ ) = ⎢ ⎥=A ⎣ 4 2 0 ⎦ ⎢ 2 0⎥ ⎣ 4 2 0 ⎦ ⎣ ⎦ he Thus (A′)′ = A (ii) We have ⎡3 3 2⎤ ⎡ 2 −1 2 ⎤ ⎡5 3 − 1 4⎤ ⇒A+B=⎢ is A= ⎢ ⎥, B = ⎢1 ⎥ ⎥ ⎣4 2 0⎦ ⎣ 2 4⎦ ⎣5 4 4⎦ bl ⎡ 5 5⎤ ⎢ ⎥ Therefore (A + B)′ = ⎢ 3 − 1 4 ⎥ ⎢ 4 4 ⎥⎦ pu ⎣ ⎡ 3 4⎤ ⎡ 2 1⎤ be T ⎢ ⎥ ⎢ ⎥ Now A′ = ⎢ 3 2 ⎥ , B′ = ⎢−1 2 ⎥ , re ⎢ 2 0⎥ ⎢⎣ 2 4 ⎦⎥ ⎣ ⎦ o R ⎡ 5 5⎤ ⎢ ⎥ tt E So A′ + B′ = ⎢ 3 −1 4 ⎥ ⎢ 4 4 ⎥⎦ ⎣ C Thus (A + B)′ = A′ + B′ (iii) We have no N ⎡ 2 −1 2 ⎤ ⎡ 2 k −k 2k ⎤ kB = k ⎢ = ⎣1 2 4 ⎥⎦ ⎢⎣ k 2k 4k ⎥⎦ © ⎡ 2k k ⎤ ⎡ 2 1⎤ ⎢ −k 2k ⎥ = k ⎢−1 2⎥ = kB′ Then (kB)′ = ⎢ ⎥ ⎢ ⎥ ⎢⎣ 2k 4k ⎥⎦ ⎢⎣ 2 4⎥⎦ Thus (kB)′ = kB′  MATRICES 85 ⎡ −2⎤ ⎢ ⎥ Example 21 If A = ⎢ 4 ⎥ , B = [1 3 −6] , verify that (AB)′ = B′A′. ⎢⎣ 5 ⎥⎦ Solution We have ⎡ −2 ⎤ he ⎢ ⎥ A = ⎢ 4 ⎥ , B = [1 3 − 6] ⎣⎢ 5 ⎦⎥ is ⎡ −2 ⎤ ⎡ −2 −6 12 ⎤ then ⎢ 4 ⎥ 1 3 −6 AB = ⎢ ⎥ [ ] = ⎢⎢ 4 12 −24⎥⎥ ⎢⎣ 5 ⎥⎦ ⎢⎣ 5 15 −30 ⎥⎦ bl ⎡ 1⎤ A′ = [–2 4 5] , B′ = ⎢ 3 ⎥ Now pu ⎢ ⎥ ⎢⎣− 6 ⎥⎦ be T ⎡ 1⎤ ⎡ −2 4 5⎤ re ⎢ 3 ⎥ −2 4 5 = ⎢ −6 12 15 ⎥⎥ = (AB)′ B′A′ = ⎢ ⎥ [ ] ⎢ o R ⎢⎣− 6 ⎥⎦ ⎢⎣12 −24 −30 ⎥⎦ tt E Clearly (AB)′ = B′A′ 3.6 Symmetric and Skew Symmetric Matrices C Definition 4 A square matrix A = [aij] is said to be symmetric if A′ = A, that is, [aij] = [aji] for all possible values of i and j. no N ⎡ 3 2 3⎤ ⎢ ⎥ For example A = ⎢ 2 −1.5 −1 ⎥ is a symmetric matrix as A′ = A ⎢ 3 −1 1 ⎥⎦ ⎣ © Definition 5 A square matrix A = [aij] is said to be skew symmetric matrix if A′ = – A, that is aji = – aij for all possible values of i and j. Now, if we put i = j, we have aii = – aii. Therefore 2aii = 0 or aii = 0 for all i’s. This means that all the diagonal elements of a skew symmetric matrix are zero.  86 MATHEMATICS ⎡ 0 e f⎤ For example, the matrix B = ⎢ −e 0 g ⎥⎥ is a skew symmetric matrix as B′= –B ⎢ ⎢⎣ − f −g 0 ⎥⎦ Now, we are going to prove some results of symmetric and skew-symmetric he matrices. Theorem 1 For any square matrix A with real number entries, A + A′ is a symmetric matrix and A – A′ is a skew symmetric matrix. Proof Let B = A + A′, then is B′ = (A + A′)′ = A′ + (A′)′ (as (A + B)′ = A′ + B′) bl = A′ + A (as (A′)′ = A) = A + A′ (as A + B = B + A) pu = B Therefore B = A + A′ is a symmetric matrix be T Now let C = A – A′ re C′ = (A – A′)′ = A′ – (A′)′ (Why?) o R = A′ – A (Why?) = – (A – A′) = – C tt E Therefore C = A – A′ is a skew symmetric matrix. Theorem 2 Any square matrix can be expressed as the sum of a symmetric and a C skew symmetric matrix. Proof Let A be a square matrix, then we can write no N 1 1 A = (A + A′) + (A − A′) 2 2 From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is © 1 a skew symmetric matrix. Since for any matrix A, (kA)′ = kA′, it follows that (A + A′) 2 1 is symmetric matrix and (A − A′) is skew symmetric matrix. Thus, any square 2 matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.  MATRICES 87 ⎡ 2 −2 −4 ⎤ Example 22 Express the matrix B = ⎢−1 3 4 ⎥ as the sum of a symmetric and a ⎢ ⎥ ⎢⎣ 1 −2 −3⎥⎦ skew symmetric matrix. Solution Here he ⎡ 2 −1 1 ⎤ B′ = ⎢⎢− 2 3 −2 ⎥⎥ ⎢⎣ −4 4 −3⎥⎦ is ⎡ −3 −3 ⎤ ⎢ 2 2 2⎥ ⎡ 4 −3 −3⎤ ⎢ ⎥ 1⎢ ⎢ −3 1 ⎥, 2⎥⎥ = ⎢ 1 bl Let P = (B + B′) = − 3 6 3 ⎥ 2 2⎢ 2 ⎢⎣ −3 2 −6 ⎥⎦ ⎢ −3 ⎥ ⎢ 1 −3 ⎥ ⎣⎢ 2 ⎦⎥ pu ⎡ −3 −3 ⎤ ⎢ 2 2 2⎥ ⎢ ⎥ −3 be T Now P′ = ⎢ 3 1 ⎥= P ⎢2 ⎥ re ⎢ ⎥ ⎢ −3 1 o R −3 ⎥ ⎣⎢ 2 ⎦⎥ 1 P = (B + B′) is a symmetric matrix. tt E Thus 2 ⎡ −1 −5 ⎤ ⎢0 2 2⎥ C ⎡0 −1 −5⎤ ⎢ ⎥ 1⎢ ⎥ Q = (B – B′) = ⎢1 0 6 ⎥ = ⎢⎢ 3⎥ 1 1 Also, let 0 2 2 2 ⎥ no N ⎢⎣5 −6 0 ⎥⎦ ⎢ ⎥ ⎢ 5 −3 0⎥ ⎣⎢ 2 ⎦⎥ ⎡ 1 5⎤ ⎢ 0 2 3⎥ © ⎢ ⎥ ⎢ −1 Then Q′ = 0 −3⎥ = − Q ⎢2 ⎥ ⎢ ⎥ ⎢ −5 3 0 ⎥ ⎣⎢ 2 ⎦⎥  88 MATHEMATICS 1 Thus Q= (B – B′) is a skew symmetric matrix. 2 ⎡ −3 −3 ⎤ ⎡ −1 −5 ⎤ ⎢2 2 2⎥ ⎢ 0 2 2 ⎥ ⎡2 ⎢ ⎥ ⎢ ⎥ −2 −4 ⎤ −3 P+Q=⎢ 1 ⎥+⎢ 3 ⎥ = ⎢⎢−1 3 4 ⎥⎥ = B 1 he Now 3 0 ⎢2 ⎥ ⎢2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ 1 −2 −3⎥⎦ ⎢ −3 1 −3 ⎥ ⎢ 5 −3 0⎥ ⎢⎣ 2 ⎥⎦ ⎢⎣ 2 ⎥⎦ is Thus, B is represented as the sum of a symmetric and a skew symmetric matrix. EXERCISE 3.3 bl 1. Find the transpose of each of the following matrices: ⎡5⎤ ⎡ −1 5 6 ⎤ pu ⎢1⎥ ⎡ 1 −1 ⎤ ⎢ ⎥ (i) ⎢ ⎥ (ii) ⎢ ⎥ (iii) ⎢ 3 5 6 ⎥ ⎢2⎥ ⎣2 3⎦ ⎢−1 ⎥ ⎢ 2 3 −1⎥ be T ⎣ ⎦ ⎣ ⎦ re ⎡ −1 2 3⎤ ⎡ −4 1 −5⎤ o R 2. If A = 5 7 9 and B = ⎢ 1 2 0 ⎥ , then verify that ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ −2 1 1 ⎥⎦ ⎢⎣ 1 3 1⎥⎦ tt E (i) (A + B)′ = A′ + B′, (ii) (A – B)′ = A′ – B′ ⎡ 3 4⎤ C ⎡ −1 2 1⎤ 3. If A′ = ⎢−1 2 ⎥ and B = ⎢ 1 2 3⎥⎦ , then verify that ⎢ ⎥ ⎢⎣ 0 1 ⎥⎦ ⎣ no N (i) (A + B)′ = A′ + B′ (ii) (A – B)′ = A′ – B′ ⎡ −2 3 ⎤ ⎡ −1 0 ⎤ 4. If A′ = ⎢ ⎥ and B = ⎢ ⎥ , then find (A + 2B)′ ⎣ 1 2⎦ ⎣ 1 2⎦ © 5. For the matrices A and B, verify that (AB)′ = B′A′, where ⎡1 ⎤ ⎡0⎤ ⎢ ⎥ , B = −1 2 1 (i) A = ⎢−4 ⎥ [ ] (ii) A = ⎢⎢1 ⎥⎥ , B = [1 5 7] ⎢⎣ 3 ⎥⎦ ⎢⎣ 2 ⎥⎦  MATRICES 89 ⎡ cos α sin α ⎤ 6. If (i) A = ⎢ ⎥ , then verify that A′ A = I ⎣ − sin α cos α ⎦ ⎡ sin α cos α ⎤ (ii) If A = ⎢ ⎥ , then verify that A′ A = I ⎣ − cos α sin α ⎦ he ⎡1 −1 5 ⎤ 7. (i) Show that the matrix A = ⎢⎢−1 2 1 ⎥⎥ is a symmetric matrix. ⎢⎣ 5 1 3⎥⎦ is ⎡ 0 1 −1 ⎤ bl (ii) Show that the matrix A = ⎢ −1 0 1 ⎥ is a skew symmetric matrix. ⎢ ⎥ pu ⎢⎣ 1 −1 0 ⎥⎦ ⎡1 5 ⎤ 8. For the matrix A = ⎢ ⎥ , verify that ⎣6 7 ⎦ be T (i) (A + A′) is a symmetric matrix re o R (ii) (A – A′) is a skew symmetric matrix ⎡ 0 a b⎤ tt E A − A′ , when A = ⎢ −a 0 c ⎥⎥ 1( ) 1( ) 9. Find A + A′ and 2 2 ⎢ ⎢⎣ −b −c 0 ⎥⎦ C 10. Express the following matrices as the sum of a symmetric and a skew symmetric matrix: no N ⎡ 6 −2 2 ⎤ ⎡ 3 5⎤ ⎢ (i) ⎢ ⎥ (ii) ⎢−2 3 −1 ⎥⎥ ⎣1 −1⎦ ⎢⎣ 2 −1 3 ⎥⎦ © ⎡ 3 3 −1⎤ ⎢−2 −2 1⎥ ⎡ 1 5⎤ (iii) ⎢ ⎥ (iv) ⎢ ⎥ ⎣ −1 2 ⎦ ⎢⎣ −4 −5 2⎥⎦  90 MATHEMATICS Choose the correct answer in the Exercises 11 and 12. 11. If A, B are symmetric matrices of same order, then AB – BA is a (A) Skew symmetric matrix (B) Symmetric matrix (C) Zero matrix (D) Identity matrix ⎡ cos α − sin α ⎤ 12. If A = ⎢ , then A + A′ = I, if the value of α is ⎣sin α cos α ⎥⎦ he π π (A) (B) 6 3 3π is (C) π (D) 2 3.7 Elementary Operation (Transformation) of a Matrix bl There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or pu transformations. (i) The interchange of any two rows or two columns. Symbolically the interchange of ith and jth rows is denoted by Ri ↔ Rj and interchange of ith and jth column is be T denoted by Ci ↔ Cj. re ⎡1 1⎤ ⎡−1 3 1⎤ o R 2 ⎢ ⎥ ⎢ ⎥ For example, applying R1 ↔ R2 to A = ⎢−1 3 1 ⎥ , we get ⎢1 2 1⎥. ⎢5 6 7 ⎥⎦ ⎢5 6 7 ⎥⎦ ⎣ ⎣ tt E (ii) The multiplication of the elements of any row or column by a non zero number. Symbolically, the multiplication of each element of the ith row by k, C where k ≠ 0 is denoted by Ri → k Ri. The corresponding column operation is denoted by Ci → kCi no N ⎡ 1⎤ 1 ⎡ 1 2 1⎤ ⎢ 1 2 7⎥ For example, applying C3 → C3 , to B = ⎢ ⎥ , we get ⎢ ⎥ 7 ⎣ −1 3 1⎦ ⎢ −1 3 1⎥ ⎢⎣ 7 ⎥⎦ © (iii) The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number. Symbolically, the addition to the elements of ith row, the corresponding elements of jth row multiplied by k is denoted by Ri → Ri + kRj.  MATRICES 91 The corresponding column operation is denoted by Ci → Ci + kCj. ⎡ 1 2⎤ ⎡1 2⎤ For example, applying R2 → R2 – 2R1, to C = ⎢ ⎥ , we get ⎢ ⎥. ⎣ 2 −1⎦ ⎣ 0 −5 ⎦ 3.8 Invertible Matrices Definition 6 If A is a square matrix of order m, and if there exists another square he matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A and it is denoted by A– 1. In that case A is said to be invertible. ⎡2 3⎤ ⎡ 2 −3 ⎤ For example, let A= ⎢ ⎥ and B = ⎢ ⎥ be two matrices. ⎣1 2 ⎦ ⎣ −1 2 ⎦ is ⎡ 2 3 ⎤ ⎡ 2 −3 ⎤ Now AB = ⎢ ⎥⎢ ⎥ ⎣ 1 2 ⎦ ⎣ −1 2 ⎦ bl ⎡ 4 − 3 −6 + 6 ⎤ ⎡ 1 0 ⎤ = ⎢ ⎥=⎢ ⎥=I ⎣ 2 − 2 −3 + 4 ⎦ ⎣ 0 1 ⎦ pu ⎡1 0 ⎤ Also BA = ⎢ ⎥ = I. Thus B is the inverse of A, in other be T ⎣0 1 ⎦ words B = A– 1 and A is inverse of B, i.e., A = B–1 re o R $ Note 1. A rectangular matrix does not possess inverse matrix, since for products BA tt E and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order. C 2. If B is the inverse of A, then A is also the inverse of B. Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. no N Proof Let A = [aij] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C. Since B is the inverse of A © AB = BA = I... (1) Since C is also the inverse of A AC = CA = I... (2) Thus B = BI = B (AC) = (BA) C = IC = C Theorem 4 If A and B are invertible matrices of the same order, then (AB)–1 = B–1 A–1.  92 MATHEMATICS Proof From the definition of inverse of a matrix, we have (AB) (AB)–1 = 1 or A–1 (AB) (AB)–1 = A –1 I (Pre multiplying both sides by A–1) or (A–1A) B (AB)–1 = A –1 (Since A–1 I = A–1) or IB (AB)–1 = A –1 he or B (AB)–1 = A –1 or B–1 B (AB)–1 = B –1 A –1 or I (AB)–1 = B –1 A –1 is Hence (AB)–1 = B –1 A –1 3.8.1 Inverse of a matrix by elementary operations bl Let X, A and B be matrices of, the same order such that X = AB. In order to apply a sequence of elementary row operations on the matrix equation X = AB, we will apply these row operations simultaneously on X and on the first matrix A of the product AB on RHS. pu Similarly, in order to apply a sequence of elementary column operations on the be T matrix equation X = AB, we will apply, these operations simultaneously on X and on the second matrix B of the product AB on RHS. re o R In view of the above discussion, we conclude that if A is a matrix such that A–1 exists, then to find A–1 using elementary row operations, write A = IA and apply a sequence of row operation on A = IA till we get, I = BA. The matrix B