Business Mathematics Part A PDF

BCOC-134 Business Mathematics and Statistics School of Management Studies PART A : BUSINESS MATHEMATICS MATRICES Unit 1 : Introduction to Matrices...

BCOC-134 Business Mathematics and Statistics School of Management Studies PART A : BUSINESS MATHEMATICS MATRICES Unit 1 : Introduction to Matrices 5 Unit 2 : Determinants 21 Unit 3 : Inverse of Matrices 35 Unit 4 : Application of Matrices in Business and 57 Economics DIFFERENTIALS CALCULUS Unit 5 : Mathematical Functions 75 Unit 6 : Limit and Continuity 100 Unit 7 : Concept of Differentiation 115 Unit 8 : Maxmima and Minimima of Functions 135 Unit 9 : Application of Derivatives 150 BASIC MATHEMATICS OF FINANCE Unit 10 : Interest Rates 175 Unit 11 : Compounding and Discounting 195 PROGRAMME DESIGN COMMITTEE B.COM (CBCS) Prof. Madhu Tyagi Prof. D.P.S. Verma (Retd.) Prof. R. K. Grover (Retd.) Former Director Department of Commerce SOMS, IGNOU, New Delhi SOMS, IGNOU University of Delhi, Delhi Faculty Members Prof. R.P. Hooda Prof. K.V. Bhanumurthy (Retd.) SOMS, IGNOU Former Vice-Chancellor Department of Commerce Prof. N V Narasimham MD University, Rohtak University of Delhi, Delhi Prof. Nawal Kishor Prof. B. R. Ananthan Prof. Kavita Sharma Former Vice-Chancellor Department of Commerce Prof. M.S.S. Raju Rani Chennamma University University of Delhi, Delhi Belgaon, Karnataka Dr. Sunil Kumar Prof. Khurshid Ahmad Batt Prof. I. V. Trivedi Dean, Faculty of Commerce & Dr. Subodh Kesharwani Former Vice-Chancellor Management Dr. Rashmi Bansal M. L. Sukhadia University, University of Kashmir, Srinagar Udaipur Dr. Madhulika P Sarkar Prof. Debabrata Mitra Prof. Purushotham Rao (Retd.) Department of Commerce Dr. Anupriya Pandey Department of Commerce University of North Bengal, Osmania University, Hyderabad Darjeeling COURSE DESIGN AND PREPARATION TEAM Prof. Madhu Tyagi Prof. G.P. Singh (Retd.) Faculty Members Former Director University of Swarashtra, SOMS, IGNOU SOMS, IGNOU Gujarat (Units 1 to 4) Prof. N V Narasimham Prof. Nawal Kishor Dr. H. K. Dogi Dr. Sarabjit Singh Kaur University of Delhi, Delhi Dayal Singh College Prof. M.S.S. Raju Units 10 to 11 University of Delhi, Delhi Dr. Sunil Kumar Units 5 to 9 Dr. Subodh Kesharwani Dr. Vidya Ratan Dr. Rashmi Bansal Sriram College of Commerce Dr. O.P. Gupta Dr. Madhulika P Sarkar University of Delhi, Delhi University of Delhi, Delhi Dr. Anupriya Pandey Prof. Brahmha Bhatt Dr. C.R. Kothari School of Commerce Rajsthan University, Course Coordinators and Gujarat University Ahemdabad Jaipur Editors Prof. M.S. Senam Raju Prof. M.S. Senam Raju Prof. (Mrs.) Sarla Achuthan SOMS, IGNOU, New Delhi SOMS, IGNOU, New Delhi Gujarat University, Ahemdabad Dr. Anupriya Pandey (Unit 15,16,17 & 18) SOMS, IGNOU, New Delhi Content Editing (Part-A) Prof. Gopinath Pradhan (Retd.) SOSS, IGNOU, New Delhi Part-B adopted from ECO-07 & MCO-03 MATERIAL PRODUCTION Mr. Y.N. Sharma Mr. Sudhir Kumar Assistant Registrar (Publication) Section Officer (Pub.) MPDD, IGNOU, New Delhi MPDD, IGNOU, New Delhi January, 2020 © Indira Gandhi National Open University, 2020 ISBN: All rights reserved. No part of this work may be reproduced in any form, by mimeograph or any other means, without permission in writing from the Indira Gandhi National Open University. Further information on the Indira Gandhi National Open University courses may be obtained from the University’s office at Maidan Garhi, New Delhi-110 068. Printed and published on behalf of the Indira Gandhi National Open University, New Delhi, by the Registrar, MPDD, IGNOU. Laser typeset by Tessa Media & Computers, C-206, A.F.E-II, Jamia Nagar, New Delhi-110025 Printed at: COURSE INTRODUCTION This is one of the core courses in B.Com programme under CBCS scheme. The main objective of this course is to familiarize the students with the application of Mathematics and Statistical techniques which will facilitate in business decision making. This course consists of two parts, viz., PART- A: Business Mathematics comprising of 11 units and PART B: Business Statistics comprising of 7 units (unit 12 to unit 18). The brief introduction of Part-A is as follows: PART A: BUSINESS MATHEMATICS This Part of the course, Business Mathematics, aims at introducing the learners to basic mathematical applications in the area of matrices, differential calculus and financial mathematics to solve simple business and economic problems. This part consists of 11 Units. MATRICES Unit 1 : Introduction to matrices discusses the concept, types of matrices, matrix algebra, Transpose of Matrix and its calculation. Unit 2: Determinants explains computation of the value of determinants, its properties minors and cofactors and application of cramer’s rule to solve system of linear equations. Unit 3: Inverse of Matrices defines the inverse matrix, its properties and computation of methods of: i) determinant and Adjoint Route, ii) Elementary Operations Route, explains inverse and Rank of a Matrix and systems of equations. Unit 4: Application of Matrices in Business and Economics discusses application and use of matrices for Business and Economic decision making. DIFFERENTIAL CALCULUS Unit 5: Mathematical Functions and Types defines functions and there types such as algebraic, transcendental, and inverse and composite functions. Its also deals with graph of some functions and functions relating to Business and Economics. Unit 6: Limit and continuity deals with the limit of a function its properties, method of factorization and properties of continuity. Unit 7: Differentiation covers the differentiation by first principle rule of differentiation, standard derivatives. It also discusses differentiation of implicit functions, using logarithms and parametric function. Unit 8: Maxima & Minima Functions explains the higher order derivatives, increasing and decreasing functions and also the function of maxima and minima. Unit 9 Applications of differentials deals with demand and supply functions, elasticity of demand and supply functions, average and marginal costs, revenue functions and profit maximization. BASIC MATHEMATICS OF FINANCE Unit 10: Interest rates discusses meaning and concept of interest, different types of interest and special cases of compound rate of interest. Unit 11: Compounding and Discounting covers the calculation of normal and effective rates of interest, present value and types of discounts. s Introduction to UNIT 1 INTRODUCTION TO MATRICES Matrices Structure 1.0 Objectives 1.1 Introduction 1.2 Matrix 1.2.1 Types of Matrices 1.3 Matrix Algebra 1.3.1 Equality of Matrices 1.3.2 Addition and subtraction of two Matrices 1.3.3 Multiplication of Matrix by a scalar quantity 1.3.4 Multiplication of Two Matrices 1. 4 Transpose of a Matrix 1.4.1 Symmetric Matrices 1.4.2 Skew Symmetric Matrices 1.4.3 Orthogonal Matrices 1.5 Let Us Sum Up 1.6 Key Words 1.7 Some Useful Books 1.8 Answer or Hints to Check Your Progress 1.9 Exercises with Answer/Hints 1.0 OBJECTIVES After going through this unit, you will be able to understand: i) Basic concept of matrix; ii) Types of the matrices; iii) Basic operations of matrix algebra; and iv) Transpose of a matrix. 1.1 INTRODUCTION Matrix (matrices in plural) is an arrangement of numbers into rows and columns. Because of its features of (i) compact notation for describing sets of data and (ii)efficient methods for manipulating data sets, it becomes a handy tool for finding solutions to problems which can be represented in linear equation system. Needless to say, matrix algebra finds wide applications covering the fields such as Engineering, Economics and Business, Sociology, Statistics, Physics, Medicine and Information Technology. For a better understanding of the applications, consider the following examples: sociologists use matrices to study the dominance within a group; demographers use these to study births and survivals, industries and businesses take the help of matrices for fast and accurate in decision making in the areas like evaluation of customers preferences to produce and sell. Some use linear programming techniques that is based on matrix formulations of data to maximise profit and thus plan production or 5  Business Mathematics availability of raw materials. Use also is made of the matrices to arrive at a decision on the location of business, marketing of the products or arranging financial resources. Economists use matrices to examine Inter-Industry flows, for studying game theory and to construct the system of social accounting. Moreover, in medical studies, scientists use data in matrix form to determine a statistically valid rate of efficacy of a drug before prescribing it in hospitals and pharmacies. Many IT companies also use matrices as data structures to track user information, perform search queries, and manage databases. Check Your Progress 1 1) What is a matrix? 2) Why would industries and businesses use matrices? 3) Matrix formulation of data that is used in linear programming is used for what purposes? 1.2 MATRIX Definition: A matrix is defined as a rectangular array of numbers arranged in rows and columns enclosed by a pair of brackets viz., [ ] or ( ). For example, the following array of numbers shows a matrix as 11 42 22 84 10 15 60 25 41 28 45 51 On the basis of number of rows and columns that a matrix has, we decide its dimension or its order. By convention, rows are expressed first while columnssecond in a matrix. Since the above matrix has 3 rows and 4 columns, we say that its dimension (or order) is 3 x 4, The numbers that appear in the rows and columns are called elements of the matrix. In the matrix above, the element in the first column of the first row is 11; the element in the second column of the first row is 42. Following the same logic, we can identify the other elements. A matrix is usually denoted by a capital letter and it’s elements by corresponding small letters with two subscripts which indicate row and column. For example, an element represented as a23 in a matrix, is read as its to be position in 2nd row and 3rd column. Thus, a matrix having m rows and n columns can be written as.. =....... 6 s The above matrix can also be written as Introduction to Matrices A = [aij}mxn where I = 1,2,3…m J=1,2,3…..n Indicating a m x n order matrix. 1.2.1 Types of Matrices We will discuss the most commonly used matrices to be able to use these in business related problems. Some other types will be taken up once we get familiar with transpose of matrix. 1) Rows Matrix: A matrix which has only one row ora matrix of order 1 x n is called row matrix. Example 1: −3 0 1 2) Columns Matrix: A matrix which has only one column ora matrix of order m x 1 is called column matrix. Example2: −2 0 2 1 3) Rectangular Matrix: A matrix is said to be rectangular if the number of rows is not equal to the number of columns. Example 3: 3 7 9 4 6 9 4) Square Matrix: A matrix in which the number of rows is equal to the number of columns is called square matrix i.e., the matrix of order m x n is a square matrix, if m = n. Example 4: 1 −2 1 −3 −3 0 5 1 2 2 1 −2 1 1 −1 2 5) Diagonal Matrix: A square matrix in which all the elements except the diagonal elements are zero is called diagonal matrix. Square Matrix A = [aij] is a diagonal matrix if aij = 0 for all I≠j Example5: 1 0 0 0 0 −3 0 0 0 0 1 0 0 0 0 2 7  Business Mathematics 6) Scalar Matrix: A diagonal matrix in which all the diagonal elements are the same is called scalar matrix. Example 6: −3 0 0 0 0 −3 0 0 0 0 −3 0 0 0 0 −3 7) Identity Matrix (Unit matrix):A scalar matrix in which all the diagonal elements are one is called unit matrix or an identity matrix. An identity matrix is denoted by capital letter I. Example 7: 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 8) Triangular Matrix: A square matrix is said to be triangular if all of its elements above the main diagonal are zero (lower triangular matrix) or all of its elements below the main diagonal are zero (upper triangular matrix). Example 8: i) Lower Triangular Matrix 1 0 0 0 3 2 0 0 5 −1 −1 0 −2 3 2 1 ii) Upper Triangular Matrix 1 3 1 −2 0 2 1 5 0 0 −1 3 0 0 0 1 9) Null or Zero Matrix: A square matrix in which all the elements are zero is called zero matrix or null matrix. It is denoted by capital letter O. Example 9: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10) Symmetric Matrix: Square Matrix A = [aij] is a Symmetric Matrix if aij = aji for all i&j. we will revisit this matrix after covering the transpose of matrix in this unit. 8 s 1 3 5 Introduction to Example10: 3 2 −1 Matrices 5 −1 −1 11) Sub Matrix: A matrix obtained by deleting some rows or columns or both of a given matrix is called sub matrix of the given matrix. Example 11: 1 3 1 5 Matrices 3 2 and are the Sub matrix of 5 −1 5 −1 1 3 5 3 2 −1 5 −1 −1 Check Your Progress 2 1) What are diagonal elements of a matrix? 2) Find the elements a21, a34, a24 and a11 in the following matrix −5 12 5 9 7 6 3 1 3 2 0 5 8 7 −4 2 Also find diagonal elements. 3) Find x and y if + 2 3 2 = 1 − 1 7 4) Classify the following matrices: −5 1 0 0 1 0 0 7 (i) 0 1 0 (ii) 3 2 0 (iii) 3 0 0 1 5 −1 −1 8 0 0 0 1 3 5 (iv) 7 6 3 1 (v) 0 0 0 (vi) 0 2 −1 0 0 0 0 0 −1 8 0 0 0 1 3 5 0 8 0 0 1 3 5 3 2 −1 (vii) (viii) (ix) 0 0 8 0 5 −1 −1 5 −1 −1 0 0 0 8 2 1 3 1.3 MATRIX ALGEBRA In this section we will discuss the basic operations of matrices. We start with the idea of matrix equality before taking up the operations of addition and multiplication. In matrix algebra, the elements are ordered numbers and 9  Business Mathematics therefore operations on them have to be done in ordered manner. It may be useful to note that while we deal with the main operations such as addition and multiplication. Other operations viz., subtraction and division are derived out of those. 1.3.1 Equality of Matrices Two matrices are equal if the following three conditions are met: i) Each matrix has the same number of rows. ii) Each matrix has the same number of columns. iii) Corresponding elements within each matrix are equal. The above conditions simply require that matrices under consideration are exactly the same. Example 12: Consider the two matrices given below: 2 2 3 = and B = 3 4 3 If A = B,then x = 3 and y = 4, since corresponding elements of equal matrices have to be equal. Further, suppose that we are given a matrix as follows: =. Then C is neither equal to A nor to B as C has three columns. Consequently, matrix C is not equal to either A or B. 1.3.2 Addition and Subtraction of two Matrices Matrices can be added or subtracted if and only if they are of the same order. The sum or difference of two (m x n) matrices is another (m x n) matrix whose elements are the sum or difference of the corresponding elements of the given matrices. For two matrices A = [aij]m x n and B = [bij]m x n A±B=C where C = [cij]m x n and cij = aij ± bij for all i & j. Example 13: 0 2 3 7 3 5 For A = and B= 2 1 4 5 −1 −3 Here 0 2 3 7 3 5 A+B= + 2 1 4 5 −1 −3 0+7 2+3 3+5 7 5 8 = = 2 + 5 1 + (−1) 4 + (−3) 7 0 1 10 s And Introduction to Matrices 0 2 3 7 3 5 A - B = - 2 1 4 5 −1 −3 0−7 2−3 3−5 −7 −1 −2 = = 2 − 5 1 − (−1) 4 − (−3) −3 2 7 Negation of a Matrix: The negation of a Matrix A is denoted by –A which is obtained by replacing all the elements of A by their negation. For example, if 1 3 5 −1 −3 −5 A= then –A = 5 −1 −1 −5 1 1 So, the subtraction of two matrices A and B can be expressed as the sum of A and the negation of matrix B. A – B = A + (-B) 1.3.3 Multiplication of Matrix by a Scalar Quantity If a Matrix is multiplied by a scalar quantity, then all the elements are multiplied by that quantity. If a Matrix A = [aij]m x n is multiplied by some scalar quantity λ then λA = λ[aij]m x n = [λaij}m x n 7 3 5 7 3 5 For example, if A = , then 3A = 3 = 5 −1 −1 5 −1 −1 21 9 15 15 −3 −3 Properties of Addition of Matrices 1) Addition of Matrices is Commutative: If A and B are two matrices of same order, then A+B=B+A 2) Addition of Matrices is Associative: If A, B and C are three matrices of same order, then (A + B) + C = A + (B + C) 3) Existence of Additive Identity: If A is a matrix and O is the null matrix of the same order as that of A, then A+O=O+A=A 4) Existence of Additive Inverse:For any Matrix, A + (-A) = (-A) + A = O The following example illustrates these properties: 1 3 5 5 1 3 6 1 −7 Let A = , B= , C= 5 −1 −1 6 −1 2 −2 0 5 0 0 0 and O =. Then 0 0 0 11  Business Mathematics 1 3 5 5 1 3 6 4 8 A+B= + = ; 5 −1 −1 6 −1 2 11 −2 1 5 1 3 1 3 5 6 4 8 B+A= + = = A + B; 6 −1 2 5 −1 −1 11 −2 1 6 4 8 6 1 −7 12 5 1 (A +B) + C = + = ; 11 −2 1 −2 0 5 9 −2 6 5 1 3 6 1 −7 11 2 −4 B+C= + = ; 6 −1 2 −2 0 5 4 −1 7 1 3 5 11 2 −4 12 5 1 A + (B+C) = + = = (A + B) 5 −1 −1 4 −1 7 9 −2 6 +C; 1 3 5 0 0 0 1 3 5 A+O= + = = A and 5 −1 −1 0 0 0 5 −1 −1 1 3 5 −1 −3 −5 0 0 0 A + (-A) = + = = O. 5 −1 −1 −5 1 1 0 0 0 1.3.4 Multiplication of Two Matrices Two matrices meet the requirement of multiplication if the number of columns of first matrix is equal to the number of rows of second matrix. If the matrix is of order i.e.,it has rows and columns, then matrix must be of order where is number of rows and is number of columns which is not necessarily equal to. Then the product is another matrix = of the order (number of rows of and number of columns of ). Let A = [aij]mxnand B = [bij]nxp be two matrices.Then the product AB is the Matrix C, where C = [cij]mxp,cij = ∑ for i =1,2,3….m & j=1,2,3,4……p where 12 s Remark: In the matrix product AB, the matrix A is called the pre-factor and Introduction to Matrices matrix B is called post-factor. Example 14: 0 1 2 1 3 5 Let A = and B = 5 2 1 5 −1 −1 −1 2 1 Here the order of matrix A is 2 x 3 and the order of matrix B is 3 x 3, so the product AB is defined. AB = Where c11 = (1x0) + (3x5) + (5x-1) = 0 + 15 – 5 = 10 c12 = (1x1) + (3x2) + (5x2) = 1 + 6 + 10 = 17 c13 = (1x2) + (3x1) + (5x1) = 2 + 3 + 5 = 10 c21 = (5x0) + (-1x5) + (-1x-1) = 0 - 5 + 1 = -4 c22 = (5x1) + (-1x2) + (-1x2) = 5 - 2 - 2 = 1 c23 = (5x2) + (-1x1) + (-1x1) = 10 - 1 - 1 = 8 10 17 10 Thus, AB = −4 1 8 Consider these matrices A and B to see whether the product BA is defined. You will find that it is not. Why? Because the number of columns in B is not equal to the number of rows in A. This shows that matrix multiplication is not commutative. For two matrices A and B, if AB and BA both are defined, then it is not necessary that they are equal. 1 3 1 2 For example, if A = and B = then 2 −1 3 4 10 14 5 1 AB = and BA = −1 0 11 5 Here AB ≠ BA For two matrices A and B if AB = O, then it is not necessary that either of A, B is a null matrix. Example 15: 1 0 0 0 Let A = and B = 0 0 0 1 0 0 Here AB = = O but neither A = O nor B = O. 0 0 Properties of Matrix Multiplication 1) Associativity: Matrix multiplication is associative. For three matrices A, B and C of order m x n,n x p and p x q respectively, (AB) C = A (BC) 13  Business Mathematics 2) Distributive over Addition: Matrix Multiplication is distributive over matrix addition. For three matrices A, B and C of order m x n, n x p and p x q respectively, A (B+C) = AB + AC 3) Identity: For any matrix A of order m x n, there is an identity matrix In of order n x n and an identity matrix Im of order m x m such that Im A = A = A In. For a square matrix A of order nxn, In A = A In = A Example 16: 1 2 1 0 1 −1 If A = , B = and C = , then show that 3 4 2 −3 0 1 (I) (AB) C= A (BC) (II) A (B+C) = AB + AC (III) AI = IA = A Solution: 1 2 1 0 5 −6 i) AB = = 3 4 2 −3 11 −12 5 −6 1 −1 5 −11 (AB) C = = 11 −12 0 1 11 −23 1 0 1 −1 1 −1 BC = = 2 −3 0 1 2 −5 1 2 1 −1 5 −11 A (BC) = = 3 4 2 −5 11 −23 Therefore, (AB) C= A (BC) 1 0 1 −1 2 −1 ii) B+C= + = 2 −3 0 1 2 −2 1 2 2 −1 6 −5 A (B+C) = = 3 4 2 −2 14 −11 1 2 1 0 5 −6 AB = = 3 4 2 −3 11 −12 1 2 1 −1 1 1 AC = = 3 4 0 1 3 1 5 −6 1 1 6 −5 AB + AC = + = 11 −12 3 1 14 −11 Therefore, A (B+C) = AB + AC 1 0 1 2 1 2 iii) IA= = =A 0 1 3 4 3 4 1 2 1 0 1 2 AI= = =A 3 4 0 1 3 4 14 Hence, AI = IA = A s Introduction to Check Your Progress 3 Matrices 1) Check the following two matrices. State if they are equal. Give reason to support your answer. 1 3 1 3 6 and 5 4 5 4 8 2) Suppose that the following two matrices are equal. What are the values of x and y? 1 3 3 = , B= 5 4 5 3) When do you say matrix multiplication is defined? 4) Explain with example the Properties of Matrix Multiplication. 5) When would you say a matrix operation is not commutative? 6) Why would you say that matrix addition is associative? 1.4 TRANSPOSE OF A MATRIX The new matrix obtained by interchanging rows and columns of the original matrix is called its transpose. Suppose we have a matrix A= of order ×. We interchanged its raw and column. The matrix thus derived is known as transpose of A and denoted by or ′. Thus, if 3 2 3 4 7 A = 4 1 , then A’ =. 2 1 −5 7 −5 For any Matrix A, AA’ and A’A are always defined but need not to be equal. In the above matrices AA’ and A’A are defined but not equal because the order of AA’ is 3 x 3 while the order of A’A is 2 x 2. Properties of Transpose of a Matrix i) (A’)’ = A ii) (kA)’ = k A’ where k is some scalar quantity. iii) (A + B)’ = A’ + B’ iv) I’ = I v) (AB)’ = B’ A’ Example17: 2 3 3 4 1 0 Let A = , B = and I = 0 1 2 1 0 1 2 0 2 3 A’ = and (A’)’ = =A 3 1 0 1 2 3 6 9 3A = 3 = 0 1 0 3 6 0 2 0 6 0 (3A)’ = and 3A’=3 = = (3A)’ 9 3 3 1 9 3 15  Business Mathematics 2 3 3 4 5 7 A+B= + = 0 1 2 1 2 2 5 2 (A + B)’ = 7 2 2 0 3 2 2 0 3 2 5 2 A’ = and B’ = so, A’ + B’ = + = = 3 1 4 1 3 1 4 1 7 2 (A + B)’ 2 3 3 4 12 11 12 2 AB = = and (AB)’ = 0 1 2 1 2 1 11 1 3 2 2 0 12 2 B’A’ = = = (AB)’ 4 1 3 1 11 1 1.4.1 Symmetric Matrix Matrix A is called symmetric matrix if A’ = A. For example, if 1 5 4 A = 5 2 −1 , 4 −1 3 1 5 4 A’ = 5 2 −1 = A. So A is a Symmetric Matrix. 4 −1 3 1.4.2 Skew Symmetric Matrix Matrix A is called skew symmetric matrix if A’ = -A. For example, if 0 5 4 A = −5 0 −1 , −4 1 0 0 −5 −4 0 5 4 A’ = 5 0 1 = - −5 0 −1 = -A. So A is a skew symmetric 4 −1 0 −4 1 0 matrix. 1.4.3 Orthogonal Matrix Matrix A is called orthogonal matrix if AA’ = A’A = I. For example, if 1 2 2 A = 2 1 −2 , −2 2 −1 1 2 −2 A’ = 2 1 2. 2 −2 −1 1 2 2 1 2 −2 Now, AA’ = 2 1 −2 2 1 2 −2 2 −1 2 −2 −1 16 s 9 0 0 1 0 0 Introduction to Matrices = 0 9 0 = 0 1 0 = I 0 0 9 0 0 1 Similarly, it also can be proved that A’A = I Check Your Progress 4 1) What do you mean by is transpose of a matrix? 2 3 4 2) You are given a matrix A=. Verify that ′ ′ = A. 1 −5 9 3) How do you get an orthogonal matrix using transpose rule? 1.5 LET US SUM UP In this unit we have discussed matrices which help find unique solution to problems when expressed in equations of linear forms. The basic operations viz., additions and multiplications have been taken up after introducing the concept of a matrix and types of matrices. We have seen that matrix defined as a rectangular array of numbers arranged in rows and columns. There are matrices such as row and column matrices, identity, diagonal, null, symmetric, rectangular, triangular and orthogonal. The unit closes with a brief discussion on transpose of matrix which means obtaining a new matrix by interchanging rows and columns of the original matrix. Some special matrices like orthogonal and skew symmetric have been discussed in this part. 1.6 KEY WORDS Diagonal Matrix:Non-zero elements only in the diagonal running from the upper left to the lower right. Equality of Matrices:Two matrices are equal if each matrix has the same number of rows, columns and corresponding elements within each are also equal. Identity matrix:A matrix usually written as I, with 1 (ones) on the main diagonal and zeros elsewhere. Lower Triangular Matrix: A special kind of square matrix with all its entries above the main diagonal aszero. Matrix Multiplication: A feasible operation when the number of columns in a first matrix is equal to number of rows in a second matrix. Matrix:A way of representing data in a rectangular array. Negation of a Matrix: Elements of a matrix with their replacement by their negation. Orthogonal Matrix:Matrix A is called orthogonal matrix if AA’ = A’A = I Rectangular Matrix:A matrix with the number of rows not equal to the number of columns. Scalar Matrix: A diagonal matrix in which all the diagonal elements are the same. Scalar: A single constant, variable, or expression. 17  Business Mathematics Skew Symmetric Matrix:Matrix A is called Skew Symmetric matrix if A’ = -A Square Matrix:A matrix in which the number of rows is equal to the number of columns. Sub Matrix:A matrix obtained by deleting some rows or columns or both of a given matrix is called sub matrix of a given matrix. Symmetric Matrix:A matrix is symmetric if it equals its own transpose. Dimension(s) or Order: The number of rows and the number of columns in a matrix. Transpose of Matrix:New matrix obtained by interchanging the rows and columns of the original. Upper Triangular Matrix:A special kind of square matrix with all its entries below the main diagonal as zero. Zero (or null) Matrix:Matrix whose elements are all zeros. 1.7 SOME USEFUL BOOKS Allen, R.G.D., “Mathematical Analysis for Economists”, London: English Language Book Society and Macmillan, 1974. Archibald, G.C., Richard G.Lipsey. “An Introduction to a Mathematical Treatment of Economics”, Delhi: All India Traveller Bookseller, 1984 Chiang, A. and Kalvin Wainwright, Fundamental Methods of Mathematical Economics (Paperback), Mac Grow Hill, 2017. Dowling, Edward,T. “Schaum’s Outline Series: Theory and Problems ofMathematics for Economists”, New York: McGraw Hill Book Company, 1986. K. Sydsaeter and P. Hammond, Mathematics for Economic Analysis, PearsonEducational Asia, Delhi, 2002. Wegner, Trevor. (2016). Applied Business Statistics: Methods and Excel- Based Applications, Juta Academic. ISBN 9781485111931 Yamane, Taro, “Mathematics for Economists: An Elementary Survey”,New Delhi: Prentice Hall of India Private Limited, 1970. 1.8 ANSWER OR HINTS TO CHECK YOUR PROGRESS Check Your Progress 1 1) Matrix is an arrangement of numbers into rows and columns. 2) Matrices are used by industries and businesses to arrive at a decision on the location of business, marketing of the products or arranging financial resources. 3) To plan production or availability of raw materials. 18 s Check Your Progress 2 Introduction to Matrices 1) Diagonal Elements:All the elements aijare called diagonal elements if i=j. The elements a11, a22, a33 …ann are diagonal elements. In the above matrices the diagonal elements are 1, 4 in matrix A and 2, 0, -2 are in matrix B and 1, 0, 1, 2 are in matrix C. 2) a21 = 7, a34 = 5, a24 = 1 and a11 = -5. Diagonal Elements are -5, 6, 0, 2. 3) x = 5, y = -2. 4) (i) Identity Matrix (ii) Lower Triangular Matrix (iii) Column Matrix (iv) Row Matrix (v) Null Matrix (vi) Upper Triangular Matrix (vii) Scalar Matrix (viii) 2 x 3 Matrix (ix) 4 x 3 Matrix Check Your Progress 3 1) These two matrices are not equal since they are not of the same dimensions. 2) =1 and =4 3) The matrix multiplication AB is defined only when the number of columns in A is equal to the number of rows in B. 4) Explain associativity, distributive and identity properties. 5) Consider matrices A and B to see whether the product BA is defined. If not, that could be due to the number of columns in B is not equal to the number of rows in A. Such a result indicates that matrix multiplication is not commutative. 6) Because (A+B) +C = A +(B+C) Check Your Progress 4 1) The new matrix obtained by interchanging rows and columns of the original matrix is called its transpose. 2 1 2 3 4 2) Get ′ = 3 −5 , then transpose it to get ′ ′ = 1 −5 9 4 9 3) Take an example such that ′ = ′ =I 1.9 EXERCISES WITH ANSWER/HINTS 1) You are told that the following two matrices are equal. What are the values of x, y, and z? 4 0 0 = 6 −2 , = 6 + 4 3 1 1 Ans.: Given that A = B, we must have all the corresponding entries equal. So, we know that a1,1 = b1,1, a1,2 = b1,2, a2,1 = b2,1, and so forth. Thus, 4 = x, –2 = y + 4 and 3 = z/3 4 0 0 Rewriting the matrices as 6 −2 = 6 + 4 , then we solve for 3 1 1 x= 4, y = –6, and z = 9. 2) Why is matrix multiplication not commutative? 19  Business Mathematics Ans.: When we do not use square matrices, we cannot even try to commute multiplied matrices as the sizes wouldn't match. But even with square matrices, we don't have commutative feature always. For example, consider case of 2×2 matrices A and B. Let = and = , + + then, AB = + + + + BA= + + It may be noted that these matrices are not be the same unless we make some very specific restrictions on the values for A and B. Since we take the rows from the first matrix and multiply by columns from the second, such a process switching the order changes the values. 2 −5 1 3 4 0 7 −6 2 3) If A = , B = and C = , −2 −1 4 5 −2 3 1 −4 11 then evaluate i) A+B ii) B–C iii) 2A + B – C 5 −1 1 −4 10 −2 Ans.: (i) (ii) 3 −3 7 4 2 −8 0 0 0 (iii) 0 0 0 4) Find the matrices A and B from the following relations 6 −6 0 3 2 5 2A – B = and 2B + A = −4 2 1 −2 1 −7 3 −2 1 0 2 2 Ans.: A = B= −2 1 −1 0 0 −3 9 1 1 5 5) If A = and B = , find the matrix X such that 3A + 5B + 4 3 7 12 2X = O −16 −14 Ans. X= −47/2 −69/2 1 2 1 0 1 −1 6) If A = , B = and C = then show that 3 4 2 −3 0 1 (i) A (B + C) = AB + AC (ii) (AB)C = A(BC) 4 2 7) If A = , find (A – 2I) (A – 3I). −1 1 1 2 8) If A = , show that products AA’ and A’A are symmetric but not 3 4 equal. 20  Determinants UNIT 2 DETERMINANTS Structure 2.0 Objectives 2.1 Introduction 2.2 Computation of Value of a Determinant 2.3 Properties of Determinants 2.4 Minors and Cofactors 2.5 Use of Cramer’s Rule in Determinants to Solve System of Linear Equations 2.6 Let Sum Up 2.7 Key Words 2.8 Some Useful Books 2.9 Answer or Hints to Check Your Progress 2.10 Exercises with Answer/Hints 2.0 OBJECTIVES After going through this unit, you will be able to understand: 1) Basic concept of Determinant 2) Difference between matrix and determinant 3) Cofactors and Minors of a Determinant 4) Application of determinant in solving a system of linear equations (Cramer’s Rule) 2.1 INTRODUCTION In the preceding unit, we have seen simple operations of matrices which can be used as a background material for obtaining solution in linear form problems. In the present unit, we extend the discussion to determinant which was discovered by Cramer during solving of system of linear equations.Note that determinant is a numerical value and can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Seen geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. We get a determinant in positive or negative value form. Check Your Progress 1 1) Distinguish between matrix and determinant 2) How Cramer discovered determinant? 3) Determinant can be computed only from which type of matrix? 21 Business Mathematics 2.2 COMPUTATION OF VALUE OF A DETERMINANT The value of determinant can be calculated by the following procedure: Consider a square matrix. For each element of its first row or first column get its submatrix. Next multiply each of the selected elements with the corresponding determinant of the submatrix. Finally, add them with alternate signs. Determinant of a Matrix whose order is one(n=1) As a base case, the value of determinant of a 1*1 matrix is the single value itself. That is to say that if the Matrix A has only one element, then the element itself is the determinant of matrix A, i.e., if A = [a], then |A| = a. For Example, if A = then |A| = 2. Determinant of a Matrix whose order is two (n=2) If the order of square matrix A is 2, then|A| = (multiplication of diagonal elements) – (multiplication of off diagonal elements), i.e., if A = , then |A| = = (a11 x a22) – (a21 x a12). 2 3 For example, if A = then |A| = (2 x 2) – (1 x 3) = 4 – 3 = 1. 1 2 Determinant of a Matrix whose order is three (n=3) Let A = , ℎ Then ( ) = = = − + ℎ ℎ ℎ. ℎ In other words, go across the first row of the matrix ,( ). Multiply each entry by the determinant of the2 × 2 matrix obtained from by crossing out the row and column containing that entry.Then we add and subtract the resulting terms, alternating signs (add the -term, subtract the -term, add the -term.) 2 1 −1 For example, if = 1 1 1 , then 1 −2 −3 1 1 1 1 1 1 |A| = +2 -1 + (-1) −2 −3 1 −3 1 −2 = 2(-3 + 2) -1(-3 - 1) – 1(-2 - 1) = -2 + 4 + 3 = 5. 22 We can use the same method to compute the determinant of a 4 × 4 matrix.In Determinants fact, the method seen above can be applied for any square matrix of any size. Note that you don't have to use the first row only; you can use any row or any column, as long as you know where to put the plus and minus signs. Thus,.. if =. then |A| = ∑ (-1)i+jaij|Mij|, Where...... Mijis the sub matrix of A obtained by striking off ithrow and jtht column. Check Your Progress 2 1) What is the value of the determinant of a matrix with a single element? 2) You are given the following matrix:. Find the value of its determinant. 3) What formula would you use to find the determinant of a square matrix of any size? 2.3 PROPERTIES OF DETERMINANTS i) If all the rows and columns of a determinant are interchanged then the value of determinant does not change, i.e.,|A| = |At| 1 6 5 1 3 −2 For example, 3 2 1 = 6 2 0 −2 0 −3 5 1 −3 ii) If any two rows/columns of a determinant are interchanged, the value of determinant remains the same but the sign is reversed. 1 6 5 −2 0 −3 For example, 3 2 1 =- 3 2 1 −2 0 −3 1 6 5 iii) If any two rows/columns of a determinant are identical, the value of determinant is zero. 1 6 5 For example, 3 2 1 = 0 3 2 1 iv) If all the elements of a row/column of determinant are multiplied by some number (say, k) then the value of the determinant is multiplied by that number. 1 6 5 2 1 2 6 2 5 For example, 3 2 1 = 56and 3 2 1 −2 0 −3 −2 0 −3 23 Business Mathematics 1 6 5 =2x 3 2 1 =2x56 =112. −2 0 −3 v) If all the elements of a row/column of a determinant are added/subtracted k-times the corresponding elements of another row/column, the value of determinant remains unchanged. 1 6 5 1 + (2 6) 6 5 For example, 3 2 1 = 56 and + (2 2) 3 2 1 = −2 0 −3 −2 + (2 0) 0 −3 13 6 5 7 2 1 = 56 −2 0 −3 vi) If all the elements of a row/column of a determinant are sum/difference of two or more elements then determinant can be expressed as the sum/difference of two or more determinants. 1 6 5 1 6 2+3 1 6 2 For example, 3 2 1 = 3 2 0+1 = 3 2 0 + −2 0 −3 −2 0 2 + (−5) −2 0 2 1 6 3 3 2 1 = - 24 + 80 = 56 −2 0 −5 Check Your Progress 3 1) You have interchanged two rows/columns of a determinant. What happens to its value? 2) John is doing some manipulation with the rows and columns of a determinant but getting the same value of it. What do you think he is doing? 3) Seeta has multiplied all the elements of the last row of the square matrix A with a constant. What value of the corresponding determinant should she get because of her action? 4) Smith made changes in a determinant so as to obtain two identical columns. What value would he get of such a determinant? 2.4 MINORS AND COFACTORS Minor of an element of a determinant Let |A| = | | be a determinant of order n. The minor of the element which exists in ithrow and jth column of the determinant |A|, is the determinant left by striking off ithrow and jth column of |A|. For a determinant of order 3, 24  st st Determinants the minor of a11 is , which is obtained by deleting 1 row and 1 column of nd Similarly the minor of a22 is which is obtained by deleting 2 row and 2nd column of Illustration: 7 5 9 Given matrix = 3 8 4 , find the minors of a32and a23. 6 2 1 7 9 The minor of a32 is = (7x4) – (9x3) = 28 – 27 = 1. 3 4 7 5 The minor of a23 is = (7x2) – (5x6) = 14 – 30 = -16. 6 2 Cofactor of an element of a determinant The cofactor of an element of a determinant is the signed minor of that element. The sign of minor is determined on the value of and , i.e., the row and column in which the element exists. The cofactor of element is denoted by. The cofactor of = (−1) minor of. For a determinant of order 3, 2 the cofactor of a11 is A11 = (-1)1+1 = (-1) = 5 Similarly, the cofactor of a32 is A32 = (-1)3+2 = (-1) = – The sign of minor is + if i + j is even and the sign of minor is – if i + j is odd. 25 Business Mathematics Illustration: 1 6 5 For the matrix = 3 2 1 , let us find out the cofactors of all the −2 0 −3 elements of the corresponding determinant of the matrix. 2 1 A11 = (-1)1+1 = (-1)1+1(2x-3 – 0x1) = -6 – 0 = -6. 0 −3 3 1 A12 = (-1)1+2 = (-1)1+2(3x-3 – 1x-2) = -(-9 + 2) = 7. −2 −3 3 2 A13 = (-1)1+3 = (-1)1+3(3x0 – 2x-2) = (0+ 4)= 4. −2 0 6 5 A21 = (-1)2+1 = (-1)2+1(6x-3 – 5x0) = -(-18 –0) = 18. 0 −3 1 5 A22 = (-1)2+2 = (-1)2+2(1x-3 – 5x-2) = (-3+10) = 7. −2 −3 1 6 A23 = (-1)2+3 = (-1)2+3(1x0 – 6x-2) = -(0+12) = -12. −2 0 6 5 A31 = (-1)3+1 = (-1)3+1(6x1 – 2x5) = 6 – 10 = -4. 2 1 1 5 A32 = (-1)3+2 = (-1)3+2(1x1 – 5x3) = -(1 – 15)= 14. 3 1 1 6 A33 = (-1)3+3 = (-1)3+3(1x2 – 6x3) = 2 – 18 = -16. 3 2 Miscellaneous Examples: 1 2 0 −1 3 −1 4 1 Example1: If A = , find |A|. −2 0 −3 3 4 3 1 2 Solution: 1 2 0 −1 3 −1 4 1 = −2 0 −3 3 4 3 1 2 −1 4 1 3 4 1 3 −1 1 |A| = 1 0 −3 3 - 2 −2 −3 3 + 0 −2 0 3 – (- 3 1 2 4 1 2 4 3 2 3 −1 4 1) −2 0 −3 4 3 1 |A| = 1M11 – 2M12 + 0M13 +1M14 −1 4 1 M11 = 0 −3 3 = (-1)(-6 – 3) -4(0 – 9) + 1(0 +9) = 9 + 36 + 9 = 54 3 1 2 26  3 4 1 Determinants M12 = −2 −3 3 = 3(-6 – 3) - 4(-4 – 12) + 1(-2 + 12) = -27 +64 +10 =47 4 1 2 3 −1 1 M13 = −2 0 3 = 3(0 – 9) +1(-4 – 12) + 1(-6 – 0) = -27 -16 - 6 = -49 4 3 2 3 −1 4 M14 = −2 0 −3 = 3(0 + 9) +1(-2 + 12) + 4(-6 – 0) = 27 + 10 - 24 = 4 3 1 13. |A| = 1M11 – 2M12 + 0M13 +1M14 |A| =1x54 – 2x47 + 0x(-49) + 1x13 = 54 – 94 + 0 + 13 = -27. 0 Example2: Evaluate 0 0 Solution: Take common a from first row, b from second row and c from third row. 0 0 = abc 0 0 0 0 Now, take common a2 from first column, b2 from second column and c2 from 0 1 1 third column. We obtain a3b3c3 1 0 1 = a3b3c3 [0(0 – 1) – 1(0 – 1) + 1(1 1 1 0 -0)] = a3b3c3[0+1+1) = 2 a3b3c3. Check Your Progress 4 1) What do you mean by a minor of a square matrix? 2) What is a cofactor of a square matrix? 3) How would you get a cofactor in a square matrix? 4) You are given following determinant: 1 −2. 4 3 Find, for all its elements, minors and cofactors. 27 Business Mathematics 2.5 USE OFCRAMER’S RULEIN DETERMINANTS TO SOLVE SYSTEM OF LINEAR EQUATIONS Cramer’s Rule: This method was given by Swiss mathematician Gabriel Cramer to solve a system of linear equations in variables using determinants. Let the system of n linear equations with n variable x1, x2, x3, x4,……xn be a11x1 + a12x2 + a13x3 + …………..+ a1nxn = b1 a21x1 + a22x2 + a23x3 + …………..+ a2nxn = b2... an1x1 + an2x2 + an3x3 + …………..+ annxn= b. The above system can be written in Matrix form AX = B.. where A=. , X = and B =.......... Let D = |A| =....... Apply the property of determinant C1→x1 C1.. x1 D =....... Apply the property of determinant C1→C1 + x2C2 + x3C3 +……..+xnCn We have + + ⋯... + + ⋯... x1 D = + + ⋯... =..... + + ⋯... 28. Determinants.. = D1...... Where D1 is the determinant of A after replacing first column by. So x1 D = D1 which provides x1 = Similarly, it can be shown that x2 D = D2 Where D2 is the determinant of A after replacing 2nd column by. Which provides x2 = similarly xn = where Dnis the determinant of A after replacing nth column by. Note: For any system of linear equations If D ≠ 0, the system has unique solution and system is consistent. If D = 0 and all D1, D2, D3, D4…… Dn are also equal to zero, then the system has infinite solutions and system is consistent. If D = 0 and at least one of D1, D2, D3, D4…… Dn is not equal to zero, then the system has no solution and system is inconsistent. Illustrations: i) Solve the following system of equations by Cramer’s Rule. x – 4y –z = 11 2x – 5y +2z = 39 -3x + 2y + z = 1 Solution: 1 −4 −1 We have D = 2 −5 2 = 1(-5 -4) + 4(2 + 6) -1(4 – 15) = -9 +32 + 11 = 34 −3 2 1 Here D ≠ 0, so solution is unique and can be obtained as x= ,y= ,z= 29 Business Mathematics 11 −4 −1 D1 = 39 −5 2 = 11(-5 – 4) + 4(39 – 2) - 1(78 + 5) = -99 + 148 - 83 = -34 1 2 1 1 11 −1 D2 = 2 39 2 = 1(39 –2) – 11(2 + 6) – 1(2 + 117) = 37 – 88 –119 = -170 −3 1 1 1 −4 11 D3 = 2 −5 39 = 1(-5 –78) + 4(2 + 117) +11(4 –15) = -83 + 476 –121 = 272 −3 2 1 So, x = = = -1, y = = = -5 and z = = =8 ii) Solve the following system of equations by Cramer’s Rule. x – 3y –8z = -10 3x + y - 4z = 0 2x + 5y + 6z = 13 Solution: 1 −3 −8 We have D = 3 1 −4 = 1(6 + 20) + 3(18 + 8) -8(15 – 2) = 26 +78 - 104 = 0 2 5 6 −10 −3 −8 D1 = 0 1 −4 = -10(6 +20) +3(0 + 52) – 8(0 – 13) =-260 + 156 + 104 = 0 13 5 6 1 −10 −8 D2 = 3 0 −4 = 1(0 + 52) +10(18 + 8) -8(39 –0) = 52 + 260 – 312 = 0 2 13 6 1 −3 −10 D3 = 3 1 0 = 1(13 –0) + 3(39 – 0) - 10(15 – 2) = 13 + 117 – 130 =0 2 5 13 Here D = D1 = D2 = D3 = 0, the system is constent with infinite solutions. II) Solve the following system of equations by Cramer’s Rule. x – 3y + 4z = 3 2x -5y +7z = 6 3x -8y + 11z = 11 Solution: 1 −3 4 We have D = 2 −5 7 = 1(-55 + 56) + 3(22-21) + 4(-16+15) = 1 +3 - 4 =0 3 −8 11 3 −3 4 D1 = 6 −5 7 =3(-55 + 56) +3(66–77) + 4(-48 + 55) = 3 – 33 + 28 = -2≠0 11 −8 11 30 Since D = 0 and D1 ≠ 0, so the system has no solution. The system is Determinants inconsistent. Check Your Progress 5 1) What is Cramer’s Rule? 2) While using Cramer’s rule, when would you say that the system of linear equations is inconsistent? 3) Give an example of system of linear equations. 4) What is the meaning of inconsistent system of linear equations? 2.6 LET US SUM UP In this unit we have discussed the solution mechanism involved in a determinant. It is seen that determinant is a numerical value and can be computed from the elements of a square matrix. We have learnt to compute the value of determinant which can be calculated by the following procedure: take a square matrix; for each element of its first row or first column we get its submatrix. Next, multiply each of the selected elements with the corresponding determinant of the submatrix. Finally, add them with alternate signs. We started value of determinant of a 1*1 matrix is the single value itself. If the order of square matrix A is 2, then |A| = (multiplication of diagonal elements) – (multiplication of off diagonal elements). In case of a square matrix of order 3 × 3we need to go across the first row of the matrix. Multiply each entry by the determinant of the 2 × 2 matrix obtained by crossing out the row and column containing that entry. Then we add and subtract the resulting terms, alternating signs (add the -term, subtract the -term, add the -term.)The method seen in case of a 3 × 3can be extended to any square matrix of any size. We have also noted that there no need to confine ourselves to use the first row or column only while computing the value of a determinant. Any row or column can be used after ascertaining its position to include the plus and minus signs. Discussing the properties of the determinant we have seen that interchanging of two rows/columns leaves the value unchanged with only its sign reversed; the value of determinant equals zero when two of its rows or columns are identical; the value of the determinant remains unchanged when all the rows and columns are interchanged. The same result is obtained by adding/subtracting all the elements of a row / column of a determinant − the corresponding elements of another row/column;the value of the determinant is -time, if we multiply all elements of a row or column with a constant. We are introduced to the concept of minor and cofactor of a square matrix to compute the value of determinant. Whereas the value obtained from the determinant of a square matrixby deleting out a row and a column corresponding to the element of a matrix is called its minor, the cofactor is defined as the signed minor. Towards the last part of the unit we have been exposed to the use of determinants to solve a system of linear equations and learnt the Cramer’s rule in that context. 31 Business Mathematics 2.7 KEY WORDS Cofactor:The signed minor. Cramer’s Rule:Methodof solving a system of linear equations in variables using determinants. Determinant:A numerical value computed from the elements of a square matrix Linear Equation System:A collection of two or more linear equations involving the same number of variables. Minor:Value obtained from the determinant of a square matrixby deleting out a row and a column corresponding to the element of a matrix. 2.8 SOME USEFUL BOOKS Allen, R.G.D., “Mathematical Analysis for Economists”, London: English Language Book Society and Macmillan, 1974. Archibald, G.C., Richard G.Lipsey. “An Introduction to a Mathematical Treatment of Economics”, Delhi: All India Traveller Bookseller, 1984 Chiang, A. and Kalvin Wainwright, Fundamental Methods of Mathematical Economics (Paperback), Mac Grow Hill, 2017. Dowling, Edward,T. “Schaum’s Outline Series: Theory and Problems of Mathematics for Economists”, New York: McGraw Hill Book Company, 1986. K. Sydsaeter and P. Hammond, Mathematics for Economic Analysis, PearsonEducational Asia, Delhi, 2002. Yamane, Taro, “Mathematics for Economists: An Elementary Survey”, New Delhi: Prentice Hall of India Private Limited, 1970. 2.9 ANSWER OR HINTS TO CHECK YOUR PROGRESS Check Your Progress 1 1) A matrix is a simply an ordered arrangement of elements in a tabular form while determinant is a single numerical value which is associated to a square matrix only. 2) During solving of system of linear equations. 3) Square matrix. Check Your Progress 2 1) The determinant of a 1×1 matrix is that number itself. 2) proceed along the first row, starting with the upper left component a. We multiply the component a by the determinant of the “submatrix” formed by ignoring a's row and column. In this case, this submatrix is 32  the 1×1 matrix consisting of d, and its determinant is just d. So the first Determinants term of the determinant is ad. Next, proceed to the second component of the first row, which is the upper right component b. Multiply b by the determinant of the submatrix formed by ignoring b's row and column, which is c. So, the next term of the determinant is bc. The total determinant is simply the first term ad minus the second term bc. 3) Let the matrix be ×. Then |A| = ∑ (-1)i+jaij|Mij|, Where Mijis the sub matrix of A obtained by striking off ithrow and jtht column. Check Your Progress 3 1) The value of determinant remains the same with its sign reversed. 2) He is interchanging (i). all the rows and columns of the determinant; and (ii) adding/subtracting all the elements of a row/column of a determinant − to the corresponding elements of another row/column. 3) She should get the value of the determinant -times of the original value. 4) He would get the value of determinant equal zero. Check Your Progress 4 1) Value obtained from the determinant of a square matrixby deleting out a row and a column corresponding to the element of a matrix. 2) The cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aijis defined by A = (–1)i+j M , where M is minor of aij. 3) A cofactor is the number one gets by removing the column and row of a designated element in a matrix. The cofactor is always preceded by a positive (+) or negative (-) sign, according as the element is in a + or − position. 4) Minor of the element aij is Mij. Here a11 = 1. So M11 = Minor of a11 = 3 M12 = Minor of the element a12 = 4 M21 = Minor of the element a21 = –2 M22 = Minor of the element a22 = 1 Next go over to cofactor of aij which is written asAij. So, A11 = (–1)1+1 M11 = (–1)2 (3) = 3 A12 = (–1)1+2 M12 = (–1)3 (4) = –4 A21 = (–1)2+1 M21 = (–1)3 (–2) = 2 A22 = (–1)2+2 M22 = (–1)4 (1) = 1. Check Your Progress 5 1) Methodof solving a system of linear equations in variables using determinants. 2) When determinant D = 0 and at least one of D1, D2, D3, D4…… Dn is not equal to zero. 33 Business Mathematics 3) Take 2 + = 5 and − + = 2. When working together, we have a system. 4) A system of linear equations is called inconsistent if it has no solutions. 2.10 EXERCISES WITH ANSWER/HINTS Solve the following system of equations using Cramer’s rule. i) 5x – 7y + z = 11; 6x – 8y - z = 15; 3x +2y - 6z = 7; ii) x +2y -2z = -7; 2x - y + z = 6; x - y - 3z = -3; iii) 6x + y -3z = 5; x +3y – 2z = 5; 2x + y +4z = 8 iv) 2x – 3y - 4z = 29; -2x + 5y – z = -15; 3x – y + 5z = -11 v) 2x - y + z = 4; x +3y + 2z = 12; 3x + 2y + 3z = 10; Answers (i) x = 1; y = -1; z = -1; (ii) x = 1; y = -2; z = 2; (iii) x = 1; y = 2; z = 1; (iv) x = 2; y = -3; z = -4; (v) No solution 34  Inverse of Matrices UNIT 3 INVERSE OF MATRICES Structure 3.0 Objectives 3.1 Introduction 3.2 Inverse Matrix 3.2.1 Definition of Inverse Matrix 3.2.2 Properties of Inverse Matrix 3.2.3 Inverting a 2 × 2 Matrix 3.2.4 Computing Inverse of Bigger Matrices 3.3 Matrix Inverse Method: Determinant and Adjoint Route 3.3.1 Adjoint of a Matrix 3.3.2 Computation of Inverse using Adjoint of a Matrix 3.4 Matrix Inverse Method: Elementary Operations Route 3.4.1 Elementary Matrix Operations 3.4.2 Computation of Inverse using Elementary Row Operations 3.5 Inverse and Rank of a Matrix 3.5.1 Rank of a Matrix 3.5.2 Linear Independence 3.5.3 Invertibility and Rank of a Matrix 3.6 Solving System of Linear Equations by Matrix Inverse 3.6.1 Systems of Equations 3.7 Let Us Sum Up 3.8 Key Words 3.9 Some Useful Books 3.10 Answer or Hints to Check Your Progress 3.11 Exercises with Answer/Hints 3.0 OBJECTIVES After going through this unit, you will be able to understand: 1) Concept of the inverse of a matrix 2) Finding Inverse of a matrix using adjoint 3) Elementary operations 4) Finding Inverse of a matrix using elementary operations 5) Rank of a matrix 3.1 INTRODUCTION In the first unit of this block under the title, ‘Introduction to Matrices’, we have seen matrix operations of addition, subtraction and multiplication. However, as it may be recalled, there was no discussion on division of a 35 Business Mathematics matrix. It is because of the underlying reason that a matrix cannot be divided. While we cannot do that, there is a related concept to work with for that purpose. It is called "inversion" of a matrix. To get an intuitive idea of inverse, it is useful to recall that a simple equation like 4 = 8 is solved if divided both the sides by 4. The result of such a move is the solution of x=2. Just note that instead of dividing by 4, we could have resorted to multiplication of ¼ in both the sides of the equation to solve the problem and the answer could have been 2. What is done is to take the help of reciprocal of 4/1 in the multiplication. Thus, reciprocal ¼ is the inversion of 4/1.We get 1 on multiplying 4/1 with its reciprocal 1/4. Matrix inversion can be thought of similar to such an operation with reciprocal numbers. 3.2 INVERSE MATRIX Let be a matrix. Then inverse of is written as. Since we cannot divide a matrix, we do not write in reciprocal form of 1/. Our search for similarity of inverse matrix with scalar number system will show that just as multiplication of 4 and its reciprocal, ¼, yields 1, our multiplication of a matrix by its inverse will give the Identity Matrix. That is, × = which would be a square matrix with 1 in the diagonal. 3.2.1 Definition of Inverse Matrix The inverse of a square × matrix , is another × matrix denoted by such that = = where Iis × identity matrix. That is, multiplying a matrix by its inverse producesan identity matrix. Note, however, that not all square matrices have an inverse matrix. If the determinant of thematrix is zero, then it will not have an inverse. Since matrix whose determinant is zero is known as singular, we stipulate the condition that only a non-singular matrix will have an inverse. Such a matrix is also called invertible matrix. 3.2.2 Properties of Inverse Matrix 1) A square matrix is invertible if and only if it is non-singular. 2) The inverse of the inverse of a matrix is matrix itself i.e., (A-1)-1 = A 3) The inverse of the transpose of a matrix is same as the transpose of its inverse i.e., (A’)-1 = (A-1)’ 4) If A and B are two invertible matrices of the same order then AB is also invertible which holds true ( ) =.. 3.2.3 Inverting a × Matrix With a view to familiarize ourselves with the idea of computation of inverse, we discuss below, as an example, a 2 × 2 matrix. Let = where a, b, c and d are numbers. Its inverse is written as 36  − Inverse of Matrices = = where( − )is the determinant − value. It needs to be not equal to zero for moving forward with process of inversion. Thus,wecould do the computation only when we have a square matrix and a non-zero determinant. 4 5 Example 1: Invert the matrix = 2 3 Solution: 4 5 3 −5 3 −5 1.5 −2.5 = = × × = =. 2 3 −2 4 −2 4 −1 2 Similar to solving single equation like 4x=8, we can solve for unknown value of a matrix X, using the inversion technique. For example, if we are given that = where matrices and are known, we can use inversion to find the matrix.To do that write =. Since = , we get = or, = (due to multiplication with identity matrix). Remember that all the conditions of inversion like is a square matrix and invertible and order of is amenable to multiplication rule are satisfied to get the required solution. 3.2.4 Computing Inverse of Bigger Matrices We have seen above the inverse of a 2x2matrix. Compared to larger matrices such as a 3x3, 4x4, …, it is easy to compute the inverse. Forlarger matrices, however, we will have to use two main methods while working out the inverse. Such methods are i) inverse of a matrix using Minors, Cofactors and Adjugate (or, Adjoint) and ii) inverse of a matrix using Elementary Row Operations. We have not included the use a computer, such as the Matrix Calculator, in the present unit which can be used for inverting a matrix. Note: Inversion of a diagonal matrix is obtained by replacing each element in the diagonal with its reciprocal. For example, take a diagonal matrix 3 0 0 =. Then =. Verify that = where is identity 0 4 0 matrix. Check Your Progress 1 1) How would you explain the idea of inversion of a matrix? 2) What is the result of multiplying a square matrix with its inverse? 3) You can apply the operation of inverse to which type of matrix? 4) Write the definition of inverse of a matrix. 5) What is a singular matrix? 6) How would you identify an invertible matrix? 37 Business Mathematics 7) How do you verify that you have correctly calculated an inverse matrix? 8) Can you perform division operation on matrix? 3.3 MATRIX INVERSE METHOD: DETERMINANT AND ADJOINT ROUTE In the following, we discuss one of the two methods mentioned above on inverting a matrix. As a background for that the concept of adjoint of a matrix which is transpose of a cofactor in a square matrix is covered first. The inverse which is obtained by dividing the adjoint with determinant is given as a second step under a separate subsection. 3.3.1 Adjoint of a Matrix To invert a matrix using first of the two methods mentioned above, it is needed to understandthe concept adjoint of a matrix. If = [aij]isa square matrix of order n, then adjoint of is defined to be transpose of matrix [Aij] of order n, where Aij is cofactor of aijin |A|. In other words, let.. =......... ( ) = transpose of......... =....... Here A11 is the cofactor of a11 in |A| A12 is the cofactor of a12 in |A| and so on. Remarks 1) If be a square matrix of order , then. ( ) = ( ). = | |. , where is an identity matrix of order. 2) ( ) = ( ). ( ) 3) ( ’) = ( ( ))’ 38 Example 2:Find the adjoint of the matrix Inverse of Matrices 1 6 5 = 3 2 1 −2 0 −3 and verify the theorem. ( ) = ( ). = | |. Solution: ( ) = transpose of 1 6 5 Given that = 3 2 1 , let us find out the cofactors of all the elements −2 0 −3 of the matrix. 2 1 A11 = (-1)1+1 = (-1)1+1(2x-3 – 0x1) = -6 – 0 = -6. 0 −3 3 1 A12 = (-1)1+2