Bsic Mthemtics for Mngement UNIT 4 MATRIX ALGEBRA AND APPLICATIONS Objectives After studying this unit, you should know the: bsic concepts of mtrix methods of representing lrge quntities of dt in mtrix form vrious opertions concerning mtrices the solution methods of simultneous liner equtions pplictions of mtrix lgebr in vrious decision models. Structure 58 4.1 Introduction 4.2 Mtrix: Definition nd Nottion 4.3 Some Specil Mtrices 4.4 Mtrix Representtion of Dt 4.5 Opertions on Mtrices 4.6 Determinnt of Squre Mtrix 4.7 Inverse of Mtrix 4.8 Solution of Liner Simultneous Equtions 4.9 Applictions of Mtrices 4.10 Summry 4.11 Key Words 4.12 Further Redings 4.1 INTRODUCTION Mtrices hve proved their usefulness in quntittive nlysis of mngeril decisions in severl disciplines like mrketing, finnce, production, personnel, economics, etc. Mny quntittive methods such s liner progrmming, gme theory, Mrkov models, input-output models nd some sttisticl models hve mtrix lgebr s their underlying theoreticl bse. All these models re built by estblishing system of liner equtions which represent the problem to be solved. The simultneous liner equtions involving more thn three vribles cnnot be solved by using "ordinry lgebr". Rel-world business problems my involve more thn three vribles, then in such cses mtrices re used to represent complex system of equtions nd lrge quntities of dt in compct form. Once the system of equtions is represented in mtrix form, they cn be solved esily nd quickly by using computer. The limittion of mtrix lgebr is tht it is pplicble only in those cses where ssumption of linerity cn be mde. The min objective of this unit is to provide (i) some bsic theoreticl mtrix opertions-ddition, subtrction, nd multipliction (ii) A procedure for solving system of liner simultneous equtions, nd (iii) few pplictions of mtrix lgebr. 4.2 MATRICES: DEFINITION AND NOTATIONS A mtrix is rectngulr rry of ordered numbers. The term ordered implies tht the position of ech number is significnt nd must be determined crefully to represent the informtion contined in the problem. These numbers (lso clled elements of the mtrix) re rrnged in rows nd columns of the rectngulr rry nd enclosed by either squre brckets, [ ]; or prentheses ( ), or by pir of double verticl line.
A mtrix consisting of m rows nd n columns is written in the following form. Mtrix Algebr nd Applictions where 11, 12,... denote the numbers (or elements) of the mtrix. The dimension (or order) of the mtrix is determined by the number of rows nd columns. Here, in the given mtrix, there re m rows nd n columns. Therefore, it is of the dimension m X n (red s m by n). In the dimension of the given mtrix the number of rows is lwys specified first nd then the number of columns. Boldfce cpitl letters such s A, B, C... re used to denote entire mtrix. The mtrix is lso sometimes represented s A = [ ij ] mxn where ii denotes the element in the ith row nd the jth element of A. Some exmples of the mtrices re The mtrix A is 2x 2 mtrix becuse it hs 2 rows nd 2 columns. Similrly, the mtrix B is 2X 3 mtrix while mtrix C is 3 X 3 mtrix. Exercise 1 Tick mrk the correct lterntive indicting the dimension of the mtrix 2 3 4 6 8 9 3 5 7 i) 3x4 ii) 4x3 iii) None of these 4.3 SOME SPECIAL MATRICES ) Squre mtrix A mtrix in which the number of rows equls the number of columns is clled squre mtrix. For exmple 2 3 4 6 8 9 3 5 7 is squre mtrix of dimension 3. The elements 2, 5 nd 1 in this mtrix re clled the digonl elements nd the digonl is clled the principl digonl. b) Digonl mtrix A squre mtrix, in which ll non-digonl elements re zero wheres digonl elements re non-zero, is clled digonl mtrix. For exmple 2 0 0 0 5 0 0 0 1 is digonl mtrix of dimension 3. 33 3 3 59
Bsic Mthemtics for Mngement c) Sclr mtrix A digonl mtrix in which ll digonl elements re equl is clled sclr mtrix. For exmple k 0 0 0 k 0 0 0 k 33 is sclr mtrix, where k is rel (or complex) number. d) Identity (or unit) mtrix A sclr mtrix in which ll digonl elements re equl to one, is clled n identity (or unit) mtrix nd is denoted by I. Following re two different identity mtrices e) 1 0 1 0 0 I 2 = ; I 3 = 0 1 0 0 1 0 0 1 2 2 3 3 An identity mtrix of dimension n is denoted by I n. It hs n elements in its digonl ech equl to 1 nd other elements re zero. The zero (or null) mtrix A mtrix is sid to be the zero mtrix if every element of it is zero. It is denoted s 0. Following re three different zero mtrices 0 0 0 0 0 0 0 ; 0 0 0 0 0 0 0 0 0 2 2 2 3 3 2 4.4 MATRIX REPRESENTATION OF DATA Before discussing the opertions on mtrices, it is necessry for you to know few situtions in which dt cn be represented in mtrix form. 1 Trnsporttion Problem The unit cost of trnsporttion of n item from ech of the two fctories to ech of the three wrehouses cn be represented in mtrix s shown below: Similrly, we cn lso construct time mtrix [t ij ], where t ij = time of trnsporttion of n item from fctory i to wrehouse j. Note tht the time of trnsporttion is independent of the mount shipped. 2 Distnce mtrix The distnce (in kms.) between given number of cities cn be represented s mtrix s shown below: 60
3 Diet mtrix The vitmin content of two types of foods nd two types of vitmins cn be represented in mtrix s shown below: Mtrix Algebr nd Applictions 4 Assignment mtrix The time required to perform three jobs by three workers cn be represented in mtrix s shown below: 5 Py-off mtrix Suppose two plyers A nd B ply coin tossing gme. If outcome (H, H) or (T, T) occurs, then plyer B loses Rs. 10 to plyer A, otherwise gins s shown in the mtrix: The minus sign with the py off mens tht plyer A pys to B. 6 Brnd Switching mtrix The proportion of users in the popultion surveyed switching to brnd j of n item in period, given tht they were using brnd i cn be represented s mtrix: Here the sum of the elements of ech row is 1 becuse these re proportions. 4.5 OPERATIONS ON MATRICES 1 Addition (or subtrction) of Mtrices The ddition (or subtrction) of two or more mtrices is possible only if these mtrices hve the sme dimensions, i.e. mtrices must hve the sme number of rows nd sme number of columns. The sum (or difference) of mtrices is obtined by dding (or subtrcting) the corresponding elements of the given mtrices. For exmple, if then 61
Bsic Mthemtics for Mngement Note tht A - B B - A, Exmple 1 1 - (-1) 3-7 2-4 A - B = = 2-0 4-8 2-4 A compny produces three types of products A, B nd C. The totl nnul sles in 000 s of units) of these products for the yers 1985 nd 1986 in the four regions is given below. For the yer 1985: For the yer 1986: Find the totl sles of three products for two yers. Solution : The totl sles of three products for two yers cn be obtined by dding the sles of two yers s shown below: Region Product Estern Western Southern Northern A 15+17=32 8+10=18 5+5=10 12+7=19 B 5+5=10 24+22=46 7+ 11=18 8+4=12 C 8+13=21 4+6=10 31+39=70 5+6=11 62 Properties of mtrix ddition If A, B nd C re ny three mtrices of sme dimension, then i) ii) iii) iv) Mtrix ddition is commuttive, i.e. A + B = B + A Mtrix ddition is ssocitive, i.e. (A + B) + C = A + (B + C) For ny mtrix A of dimension m X n, there is zero mtrix of the sme dimension such tht A + 0 = 0 + A = A This shows tht zero mtrix is the dditive identity If for ny mtrix A of dimension m X n, there exists nother mtrix B of the sme dimension such tht A + B = B + A = O then B is clled the dditive inverse (or negtive) of A nd is denoted by - A.
Exercise 2 If mtrices A nd B re defined s 0 2 3 7 6 3 A = ; B = 2 1 4 1 4 5 then compute ) A + B b) A - B c) B - A 2 Sclr Multipliction If A [ ij ] is ny mtrix of dimension m x n nd k is ny sclr (rel number), then the multipliction KA is obtined by simply multiplying ech element of A by the sclr K. Tht is AK = KA = [k ij ] Exmple 2 The sles figures in Exmple 1 re given in thousnds of units. If we wnt to express sles figures in ctul units, then we hve to multiply the given mtrices by 1000. For illustrtion, let us consider the dt mtrix of 1985. Tht is, if A = 15 8 5 12 5 24 7 8 8 4 31 6 then Mtrix Algebr nd Applictions Properties of sclr multipliction i) K(A + B) = KA ± KB Where A nd B re two mtrices of sme dimension nd K is sclr number. ii) (K 1 + K 2 ) A = K 1 A + K 2 A Where A is mtrix nd K 1 nd K 2 re two distinct sclr numbers. Exercise 3 If two mtrices A nd B re defined s 0 2 3 7 6 3 A = ; B = 2 1 4 1 4 5 then compute 2A + 3B. 3 Multipliction of Mtrices The mtrix multipliction consists of the following steps: ) Check on comptibility: The following dimensionl rrngement must hold for comptibility in mtrix multipliction: dimensions: led mtrix X lg mtrix = product (m x p) X (p x n) = m x n In other words, the number of columns in the first mtrix must be equl to the number of rows in the second mtrix. If this condition does not exist, then the mtrices re sid to be incomptible nd their multipliction is not defined. b) The opertion of multipliction: For multipliction of two mtrices the following procedure should be dopted: i) The element of row of the led mtrix A should be multiplied by the corresponding elements of column of the lg mtrix B. 63
Bsic Mthemtics for Mngement ii) The product is then summed nd the loction of this resulting element in the new mtrix C determines which row from A hs to be multiplied with which column from B. To illustrte this, let us tke two mtrices A nd B s defined below: then 2 3 5 2 3 A = ; B = 3 5 3 5 7 5 7 2 3 3 2 Exmple 3 There re two fmilies A nd B. There re 2 men, 3 women nd 1 child in fmily A nd 1 mn, 1 womn nd 2 children in fmily B. The recommended dily llownce for clories is; mn, 2400; womn, 1900, child, 1800, nd for proteins: mn, 55 gm; womn, 45 gm nd child, 33 gm. Represent the bove informtion by mtrices. Using mtrix multipliction, clculte the totl requirement of clories nd proteins for ech of the two fmilies. Solution: nd If you look t the dimensions of two mtrices C nd D, then you will find tht the condition for multipliction is stisfied. Therefore, the totl requirement of clories nd proteins for ech of the two fmilies is determined by multiplying C nd D, s shown below: 64 Exercise 4 1 If two mtrices of dimension m x n nd n x p re multiplied, then the resulting mtrix is of dimension: (i) m x n (ii) n x p (iii) m x p (iv) None of these
2 If A nd B re two non-zero comptible mtrices with respect to multipliction, then their product i) is lwys zero mtrix ii) is never zero mtrix iii) my be zero mtrix iv) None of these 3 A fctory employs 50 skilled workers nd 20 unskilled workers. The dily wges to skilled nd unskilled workers re Rs. 30 nd Rs. 17 respectively. Using mtrix nottion find ) b) the number of workers mtrix the totl dily pyment mde to the workers. Properties of mtrix multipliction i) Mtrix multipliction, in generl, is not commuttive. i.e., AB BA ii) Mtrix multipliction is ssocitive, i.e. A(BC) = (AB)C where A, B, C re ny three mtrices of dimension m x n, n x p, p x q respectively iii) Mtrix multipliction is distributive A (B + C) = AB + AC where A, B, C re ny three m x n, n x p nd n x p mtrices respectively. 4 Trnspose of Mtrix Let A be ny mtrix. The mtrix obtined by interchnging rows nd columns of A is clled the trnspose of A nd is denoted by A' or A t. Thus if A = [ ij ] is n m x n mtrix, then A t = [ ji ] will be n X m mtrix. For exmple, the trnspose of the mtrix is 2 3 4 A = 1 2 0 2 1 4 0 t A = 3 2 3 2 Properties of trnspose of mtrices i) Trnspose of sum (or difference) of two mtrices is the sum (or difference) of the trnsposes, i.e. (A ± B) t = A t ± B t ii) Trnspose of trnspose is the originl mtrix iii) (A t ) t = A 2 1 2 2 2 A = ; B = 1 4 2 4 0 2 0 then verify tht (AB) t = B t A t 4.6 DETERMINANT OF A SQUARE MATRIX 2 3 Trnspose of product of two mtrices is the product of their trnsposes tken in reverse order (AB) t = B t A t Exercise 5 If two mtrices A nd B re defined s The determinnt of squre mtrix is sclr (i.e. number). Determinnts re possible only for squre mtrices. For more clrity, we shll be defining it in stges, strting with squre mtrix of order 1, then for mtrix of order 2, etc. The determinnt of squre mtrix A is denoted either by A or det. A. Mtrix Algebr nd Applictions 65
Bsic Mthemtics for Mngement i) Determinnt of order 1. Let A = ( 11 ) be mtrix of order 1. Then det. A = 11 ii) Determinnt of order 2. Let 11 12 A = 21 22 be squre mtrix of order 2, then det. A is defined s 11 12 de t. A = = 11 22 2112 21 22 For exmple 3 4 det. A = = 3 2 1 4 = 2 1 2 To write the expnsion of determinnt to mtrices of order 3, 4,..., let us first define two importnt terms: ) Minor: Let A be squre mtrix of order m. Then minor of n element ij is the determinnt of the residul mtrix (or submtrix) obtined from A by deleting row i nd column j contining the element ij. In the A, the minor of the element ij is denoted by M ij. Thus, in the determinnt of order 3 11 12 13 21 22 23 31 32 33 the minor of the element 11 is obtined by deleting first row nd first column contining element 11 nd is written s 22 23 M 11 = 32 33 Similrly, minor of 12 is 21 23 M 12 = 31 33 c) Cofctor: The cofctor c ij of n element ij is defined s c ij = ( - 1) i+j M ij where M ij is the minor of n element ij. Now using the concept of minor nd cofctor, you cn write the expnsion of determinnt of order 3 s shown below: 66 The expnsion of the given determinnt cn lso be done by choosing elements ny row nd column. In the bove exmple expnsion ws done by using the elements of the first row.
Exmple 4 Find the vlue of the determinnt Mtrix Algebr nd Applictions 1 18 72 det. A = 2 40 96 2 45 75 Solution: If you expnd the determinnt by using the elements of the first column, then you will get Properties of determinnts Following re the useful properties of determinnts of ny order. These properties re very useful in expnding the determinnts. 1 The vlue of determinnt remins unchnged. If rows re chnged into columns t nd columns into rows, i.e. A = A 2 If two rows (or columns) of determinnt re interchnged, then the vlue of the determinnt so obtined is the negtive of the originl determinnt. 3 If ech element in ny row or column of determinnt is multiplied by constnt number sy K, then the determinnt so obtined is K times the originl determinnt. 4 The vlue of determinnt in which two rows (or columns) re equl is zero. 5 If ny row or column) of determinnt is replced by the sum of the row nd liner combintion of other rows (or columns), then the vlue of the determinnt so obtined is equl to the vlue of the originl determinnt. 6 The rows (or columns) of determinnt re sid to be linerly dependent if A = 0, otherwise independent. Exmple 5 Verify the following result Applying row opertions (Property 5) 67
Bsic Mthemtics for Mngement On the given determinnt, the determinnt so obtined Expnding the new determinnt by the elements of first column, you will get Agin performing row opertions You will hve Exercise 6 If + b + c = 0, then verify the following result. 68 4.7 INVERSE OF A MATRIX If for given squre mtrix A, nother squre mtrix B of the sme order is obtined such tht AB = BA = I then mtrix B is clled the inverse of A nd is denoted by B = A - 1. Before strt discussing the procedure of finding the inverse of mtrix, it is importnt to know the following results: 1 The mtrix B = A - 1 is sid to be the inverse of mtrix A if nd only if AA - 1 = A - 1 A = I. 2 Tht is, if the inverse of squre mtrix multiplied by the originl mtrix, then result is n identity mtrix. The inverse A - 1 does not men 1/A or I/A. This is simply nottion to denote the inverse of A. 3 Every squre mtrix my not hve n inverse. For exmple, zero mtrix hs no inverse. Becuse, inverse of squre mtrix exists only if the vlue of its determinnt is non-zero, i.e. A - 1 exists if nd only if A 0. For exmple, let B be the inverse of the mtrix A, then AB = BA = I or AB = I or A. B = 1 ( I = 1) Hence A 0 4 If squre mtrix A hs n inverse, then it is unique. It cn lso be proved by letting two inverses B nd C of A. We then hve AB = BA = I... (i) nd AC = CA = I...(ii) Pre-multiplying (i) by C, we get CAB = CI or
or i) ii) iii) iv) IB = CI B = C(CA = I) his implies tht the inverse of squre mtrix is unique. Singulr Mtrix A mtrix is sid to be singulr if its determinnt is equl to zero; Otherwise nonsingulr. Properties of the inverse The inverse of the inverse is the originl mtrix, i.e. The inverse of the trnspose of mtrix is the trnspose of its inverse, i.e. (A t ) -1 = (A -1 ) t The identity mtrix is its own inverse, i.e. I -1 = I The inverse of the product of two non-singulr mtrices is equl to the product of two inverses in the reverse order, i.e. (AB) -1 = B -1. A -1 Method of finding inverse of mtrix The procedure of finding inverse of squre mtrix A = [ ij ] of order n cn be summrised in the following steps : Mtrix Algebr nd Applictions 1 Construct the mtrix of co-fctors of ech element ij in A s follows: In this cse cofctors re the elements of the mtrix 2 Tke the trnspose of the mtrix of cofctors constructed in step 1. It is clled djoint of A nd is denoted by Adj. A. 3 Find the vlue of A 4 Apply the following formul to clculte the inverse of A Exmple 6 Find the inverse of the mtrix Solution: The determinnt of mtrix A is expnded with respect to the elements of first row: 69
Bsic Mthemtics for Mngement Since A 0, therefore the inverse of A exists. The mtrix of cofctor of elements A is: The dj. A is now constructed by tking trnspose of the cofctor mtrix: 9-12 9 Adj. A = (Co-fctor A) t 11 4-3 -5 2 9 Hence 70 Exercise 7 For the mtrix A = i) Clculte A -1 ii) Verify (A t ) -1 = (A -1 ) t 1 4 0-1 2 0 0 0 2 iii) Verify (dj A) -1 = dj (A -1 ) 4.8 SOLUTION OF LINEAR SIMULTANEOUS EQUATIONS As mentioned erlier in this unit, mtrix lgebr is useful in solving set of liner simultneous equtions involving more thn two vribles. Now the procedure for getting the solution will be demonstrted.
Consider the set of liner simultneous equtions 2x + 5y - 2z = 3 These equtions cn lso be solved by using ordinry lgebr. However, to demonstrit the use of mtrix lgebr, the first step is to write the given system of equtions mtrix form s follows: Mtrix Algebr nd Applictions or Where AX = B is known s the coefficient mtrix in which coefficients of x re written in first column, coefficients of y in second column nd the coefficients of z in the third column. is the mtrix of unknown vribles x, y nd z, nd is the mtrix formed with the right hnd terms in equtions which do not involve unknowns x, y nd z. Generlising the sitution, let us consider m liner equtions in n-unknowns x 1, x 2,,x n ; Writing this system of equtions in mtrix form, where AX = B 71
Bsic Mthemtics for Mngement Clssifiction of liner Equtions If mtrix 13 is zero mtrix, i.e. B = 0, then the system AX = 0 is sid to be homogeneous system. Otherwise, the system is sid to be non-homogeneous. Homogeneous Liner Equtions When the system is homogeneous, i.e. b 1 = b 2 =... = b m = 0, the only possible solution is X = 0 or x 1 = x 2 =..x n = 0. It is clled trivil solution. Any other solution if it exists is clled non-trivil solution of the homogeneous liner equtions. In order to solve the eqution AX = 0, we perform such n elementry opertions or trnsformtions on the given coefficient mtrix A which does not chnge the order of the mtrix. An elementry opertion is of ny one of the following three types: i) ii) iii) The interchnge of ny two rows (or columns) The multipliction (or division) of the elements of ny row (or column) by ny non-zero number, e.g. the R i (row i) cn be replced by KR i ( K ). 0 The ddition of the elements of ny row (or column) to the corresponding elements of ny other row (or column) multiplied by ny number, e.g. R i (row i) cn be replced by R i + KR j where R j is the row j nd K 0. The elementry opertion is clled row opertion if it pplies to rows, nd column opertion if it pplies to columns. For the purpose of pplying these elementry opertions, we form nother mtrix clled ugmented mtrix s shown below: Solution Method We shll pply Guss-Jordon Method (lso clled Tringulr form Reduction Method) to solve homogeneous liner equtions. In this method the given system of liner equtions is reduced to n equivlent simpler system (i.e. system hving the sme solution s the given one). The new system looks like: x 1 +b 1 x 2 +C1x 3 = d 1 x 2 + C 2 x 3 = d 2 x 3 = d 3 This Method helps, not only to find solution to homogeneous equtions but lso to non-homogeneous system of equtions hving ny number of unknowns. Exmple 7 Solve the following system of equtions using Guss Jordon method x 1 + 3x 2 - x 3 = 0 2x 1 - x 2 + 4x 3 = 0 72 x 1-11x 2 + 14x 3 = 0 Solution: The given system of equtions in mtrix form is:
Mtrix Algebr nd Applictions The ugmented mtrix becomes Applying elementry row opertions R R 2R The new equivlent mtrix is: 2 2 R R R 3 3 1 1 Agin pplying R 3 R 3 2R2. The new equivlent mtrix is: The equtions equivlent to the given system of equtions obtined by elementry row opertions re: The lst eqution, though true, is redundnt nd the system is equivlent to x 1 + 3x 2-2x 3 = 0 x 2 - (8/7)x 3 = 0 This is not in tringulr form becuse the number of equtions being less thn the number of unknowns. This system cn be solved in terms of x 3 by ssigning n rbitrry constnt vlue, k to it. The generl solution to the given system is given by Exercise 8 Solve the following system of equtions using Guss-Jordon Method i) 4x 1 + x 2 = 0-8x 1 + 2x 2 = 0 ii) x 1-2x 2 + 3x 3 = 0 2x 1 + 5x 2 + 6x 3 = 0 Non-homogeneous Liner Equtions The non-homogeneous liner equtions cn be solved by ny of the following three methods 1 Mtrix Inverse Method 2 Crmer's Method 3 Guss-Jordon Method Agin, for the purpose of demonstrting bove solution methods, we shll consider three equtions with three unknowns. 1 Mtrix Inverse Method Let AX = B be the given system of liner equtions, nd lso A -1 be the inverse of A. Premultiplying both sides of the eqution by A -1, A -1 (AX) = A -1 B (A -1 A)X = A -1 B IX = A -1 B 73
Bsic Mthemtics for Mngement X where I is the identity mtrix. = A -1 B The vlue of X gives the generl solution to the given set of simultneous equtions. This solution is thus obtined by (i) first finding A -1, nd (ii) post multiplying A -1 by B. When the system hs solution, it is sid to be consistent, otherwise inconsistent. A consistent system hs either just one solution or infinitely mny solutions. Exmple 8 The dily cost, C of operting hospitl, is liner function of the number of inptients I, nd out-ptients, P, plus fixed cost, i.e., C = + b P + di. Given the following dt for three dys, find the, vlues of, b, nd d by setting up liner system of equtions nd using the mtrix inverse. Dy Cost No. of No. of (in Rs.) in-ptients, I out-ptients, P 1 6,950 40 10 2 6,725 35 9 3 7,100 40 12 Solution:. Bsed on the given dily cost eqution, the system of equtions for three dys cost cn be written s : + 10b + 40d = 6,950 + 9b + 35d = 6,725 + 12b + 40d = 7,100 This system cn be written in the mtrix form s follows: Which is of the form AX = B, where The inverse of mtrix A is obtined s follows: 74
Mtrix Algebr nd Applictions Since A 0, therefore inverse of mtrix A exists nd is computed s Exercise 9 A slesmn hs the following record of sles during three months for three items A, B nd C, which hve different rtes of commission. Find out the rtes of commission on items A,'B nd C. 2 Crmer's Method When the number of equtions is equl to the number of unknowns nd the determinnt of the coefficients hs non-zero vlue, then the system hs unique solution which cn be found by using Crmer's formul. D j x j =, j = 1, 2,...,n D where D = nd determinnt D j is obtined from D by replcing column j by the ij column of constnt terms (i.e. mtrix B). Exmple 9 An utomobile compny uses three types of steel, S 1, S 2 nd S 3 for producing three different types of crs C 1, C 2 nd C 3. Steel requirements (in tons) for ech type of cr nd totl vilble steel of ll the three types is summrised in the following tble. 75
Bsic Mthemtics for Mngement Determine the number of crs of ech type which cn be produced. Solution: Let x 1, x 2 nd x 3 be the number of crs of the type C 1, C 2 nd C 3 respectively which cn be produced. Then system of three liner equtions is: 2x 1 + 3x 2 + 4x 3 = 29 x 1 + x 2 + 2x 3 = 13 3x 1 +2x 2 +x 3 = 16 These equtions cn lso be represented in mtrix form s shown below: The determinnt of the coefficients mtrix is Applying Crmer's Method Hence, the number of crs of type C 1, C 2 nd C 3 which cn be produced re 2, 3 nd 4 respectively. Exercise 10 A firm mkes two products A nd B. Ech product requires production time in ech of two deprtments I nd II s shown below: 76 Totl time vilble is 80 hours nd 60 hours in deprtment I nd II respectively. Determine the number of units of product A nd B which should be produced. 4.9 APPLICATIONS OF MATRICES 1 Mrkov Models A prticulr mthemticl model which is concerned with the brnd-switching behviour of consumers who re essentilly repet-buyers of the product, is known s Mrkov brnd-switching model. These models help in predicting the mrket shre of product t time period t, if the mrket shre t the time period (t- 1) is known.
Mrkov models hve lso been used in the study of (i) equipment mintennce nd filure probbility. (ii) stock mrket price movements, etc. The generl expression for forecsting the buying levels t time t = n + 1 is given by Mtrix Algebr nd Applictions is the mtrix of trnsition probbilities. Ech element of it represents the probbility tht customer will chnge his liking from one brnd to nother in his next purchse. This is the reson for clling them trnsition probbilities nd, p ij = 1, R = mtrix of order (1 X n) representing the buying levels (or stte probbilities) t prticulr time period If we know the buying levels t time t = 0, then we cn find them t ny time by solving the bove eqution by the reltion. n j Now s the time psses, i.e. n the purchsing levels (or mrket shres) tends to settle down to n equilibrium (or stedy stte). Tht is, once n equilibrium stte is reched there will be no chnge in the future mrket shres. Thus Lt. R = Lt. R.P n n+1 n n or R = RP This reltionship cn be used to determine mrket shres in the long run. Exmple 10 Consider the following mtrix of trnsition probbilities of product vilble in the mrket in two brnds: Determine the mrket shres of ech of the brnd in equilibrium position. Solution: If the row vector (mtrix hving only one row) represents the mrket shre of the two brnds t equilibrium, then R = RP i.e. These re two liner homogeneous simultneous equtions. But these re not independent since one cn be derived from the other. Hence, in order to solve, one more eqution is needed, which is r 1 + r 2 = 1.. (iii) This is becuse the mrket shres hve been expressed in percentge, so the sum of mrket shres will be 1. Solving equtions (i) nd (ii) with the help of eqution (iii), to get mrket shres in n 77
Bsic Mthemtics for equilibrium condition, Mngement r 1 = 0.75 nd r 2 = 0.25 Hence the expected mrket shres in n equilibrium condition for brnd A will be 0.75 nd tht of brnd B will be 0.25. Exercise 11 The purchse ptterns of two brnds of toothpste cn be expressed s Mrkov process with the following trnsition probbilities Formul A Formul B Formul A 0.90 0.10 Formul B 0.05 0.95 Wht re the projected mrket shres for the two formul? 2 Input-Output Anlysis The method of "input-output nlysis" ws first proposed by Wssily W. Leotief in the 1930s. This method is bsed on the concept of "economic inter-dependence", which mens tht every sector (or industry) of the economy is relted to every other sector. Tht is, they re ll inter-dependent nd inter-relted. This mens, ny chnge in one sector (such s strike) will ffect ll other industries to vrying degree. However, this technique does not explin or estblish s to why such effects occur. The input-output model is bsed on the following ssumptions i) An economy is decomposed into n sectors (or industries), nd ech of these produces only one kind of product. Ech of the sectors uses s input, the output of the other sectors. Let x j (j =1, 2,..., n) be the gross production (output) of the jth sector. ii) Let ij represents rupee vlue of the output from sector i which sector j must consume to produce one rupee worth of its own product. It cn be clculted s follows: Rupee vlue of the product of sector i required by sector j ij = Rupee vlue of the totl output of sector j The ij 's for ll i nd j cn be represented in mtrix form s shown below: The mtrix A is the technicl input-output coefficient mtrix. This mtrix remins unchnged so long s the structure of the economy remins unchnged. iii) There is neither shortges or surpluses of product under considertion. In other words, gross product of ech sector is sufficient to meet the finl demnd s well s demnds of other sectors. Let d j (j =1, 2,..., n) be the finl demnd (in rupee vlue) for product produced by ech of n sectors. The input-output tble displyed in the following tble summrises informtion bout the economy in question. 78 If the economy is ssumed to be in stte of dynmic equilibrium (i.e. neither shortges nor surpluses) so tht the totl output is just sufficient to meet the input needs of ech sector s well s the needs of the finl demnd of ll sectors
themselves, then Output = Input = Need of ech sector + Finl demnd n x = x + d ; i ij j j=1 In mtrix nottion, we hve i for sector i = 1, 2,, n Mtrix Algebr nd Applictions The bove eqution cn lso be rewritten s: X = AX + D IX = AX + D IX AX = D (I - A)X = D (I - A)X = D X(I - A) -1 D; provided I - A 0 where I is the identity mtrix. The vlue of X gives how much ech sector must produce which is just sufficient to meet the finl demnd s well s the demnd of ll sectors themselves. Exmple 11 Given the following input-output tble, clculte the gross output so s to meet the finl demnd of 200 units of Agriculture nd 800 units of Industry. Solution: Using the nottions s discussed bove Rupee vlue of the product of sector Agriculture used by Agriculture 11 = Rupee vlue of the totl output of sector Agriculture = 300 = 0.3 1000 Similrly 600 12 = = 0.6 2000 400 21 = = 0.4 1000 1200 22 = = 0.6 2000 Thus the technologicl mtrix A nd finl demnd mtrix D, becomes 79
Bsic Mthemtics for Mngement Hence, the gross output of Agriculture nd Industry must be 2000 units nd 4000 units respectively. Exercise 12 In n economy there re two sectors A nd B nd the following tble gives the supply nd demnd position of these in million rupees: Determine the totl output, if the demnd chnges to 12 for A nd 18 for B. 4.10 SUMMARY Mtrices ply n importnt role in quntittive nlysis of mngeril decisions. They lso provide very convenient nd compct methods of writing system of liner simultneous equtions nd methods of solving them. These tools hve lso become very useful in ll functionl res of mngement. Another distinct dvntge of mtrices is tht once the system of equtions cn be set- up in mtrix form, they cn be solved quickly using computer. A number of bsic mtrix opertions (such s mtrix ddition, subtrction, multipliction) were discussed in this unit. This ws followed by discussion on mtrix inversion nd procedure for finding mtrix inverse. Number of exmples were given in support of the bove sid opertions nd inverse of mtrix. Finlly, two importnt pplictions of mtrix lgebr-predicting mrket shres using Mrkov models nd predicting the effect of chnge in the output (or demnd) of one sector of the economy on the output of the other sectors, using input-output models were discussed. 4.11 KEY WORDS Co-fctor: The number i+j C = (-1) M is clled the co-fctor of element ij in A. ij ij 80
Determinnt: A unique sclr quntity ssocited with ech squre mtrix. Identity mtrix: A mtrix in which digonl elements re equl to 1 nd ll other elements re zero. Mtrix Algebr nd Applictions Mtrix: It is n rry of numbers, rrnged in rows nd columns. Minor: The minor of n element is the determinnt of the submtrix obtined from given mtrix by deleting the row nd the column contining tht element nd is denoted by M ij. Nullmtrix: A mtrix in which ll elements re zero. Trnspose mtrix: A new mtrix obtined by interchnging rows nd columns of the originl mtrix. 4.12 FURTHER READINGS Budnicks, F.S., 1983, Applied Mthemtics for Business, Economics nd Socil Sciences, McGrw-Hill: New York. Hughes, A.J., 1983Applied Mthemtics for Business Economics, nd Socil Sciences, Irwin: Homewood. Rghwchri, M., 1985, Mthemtics for Mngement: An Introduction, Tt McGrw Hill (Indi): Delhi Weber, J.E., 1982. Mthemticl Anlysis: Business nd Economics Applictions, Hrper & Row: New York 81