Matrix Algebra and Applications - UTEP

1 3 Matrix Algebra and Addition andScalar ModelsKey ConceptsReview ExercisesCase Study ExercisesTechnology GuidesCASE STUDYThe Japanese EconomyA senator walks into your cubicle in the Congressional Budget Office. Look here, she says, I don t see why the Japanese trade representative is getting so upset with my proposal to cutdown on our use of Japanese finance and insurance. He claims that it ll hurt Japan s miningoperations. But just look at Japan s input-output table. The finance sector doesn t use anyinput from the mining sector. How can our cutting down demand for finance and insurancehurt mining? How should you respond?173 Jose Fuste Raga/Zefa/CorbisOnline you will find: Section by section tutorials A detailed chapter summary A true/false quiz Additional review exercises A Matrix Algebra tool, game theoryutility, and other 7/17/06 4:24 PM Page 173 IntroductionWe used matrices in Chapter 2 simply to organize our work. It is time we examined themas interesting objects in their own right.

2 There is much that we can do with matricesbesides row operations: We can add, subtract, multiply, and even, in a sense, divide matrices. We use these operations to study game theory and input-output models in thischapter, and Markov chains in a later calculators, electronic spreadsheets, and other computer programs can dothese Matrix operations, which is a big help in doing calculations. However, we need toknow how these operations are defined to see why they are useful and to understandwhich to use in any particular 3 Matrix Algebra and ApplicationsquickExamplesMatrix, Dimension, and EntriesAn m nmatrixAis a rectangular array of real numbers with mrows and refer to mand nas the dimensionsof the Matrix . The numbers that appear in the ma-trix are called its customarily use capital letters A, B, C, ..for the names 20133 22 0 is a 2 3matrix because it has 2 rows and 3 2310 44 1383 is a 4 2matrix because it has 4 rows and 2 entries of Aare 2, 0, 1, 33, 22, and 0.

3 The entries of Bare the numbers 2, 3, 10,44, 1, 3, 8, and Addition and Scalar MultiplicationLet s start by formally defining what a Matrix is and introducing some basic :Remember that the number of rows is given first and the number of columns easy way to remember this is to think of the acronym RC for Row then Column. quickExampleReferring to the Entries of a MatrixThere is a systematic way of referring to particular entries in a Matrix . If iand jare num-bers, then the entry in the ith row and jth column of the Matrix Ais called the ijth entryof A. We usually write this entry as aijor Aij. (If the Matrix was called B, we wouldwrite its ijth entry as bijor Bij.) Notice that this follows the RC convention: The rownumber is specified first and the column number A= 20133 22 0 ,a13=1 First row, third columna21=33 Second row, first 7/17/06 4:24 PM Page Matrix Addition and Scalar Multiplication175 According to the labeling convention, the entries of the Matrix Aabove are A= a11a12a13a21a22a23 In general, the m nmatrix Ahas its entries labeled as follows:A= We say that two matrices Aand Bare equalif they have the same dimensions andthe corresponding entries are equal.

4 Note that a 3 4matrix can never equal a 3 5matrix because they do not have the same TechnologySee the Technology Guides atthe end of the chapter to seehow matrices are entered andused in a TI-83/84or authors web-based utility,follow:Chapter 3 Tools Matrix Algebra ToolThere you will find a computa-tional tool that allows you to domatrix Algebra . Use the followingformat to enter the Matrix Aonthe previous page (spaces areoptional):A=[2, 0,133, 22, 0]To display the Matrix A, type Ainthe formula box and press Compute. Example1 Matrix EqualityLet A= 79x0 1y+1 and B= 7900 111 .Find the values of xand ysuchthat A= the two matrices to be equal, we must have corresponding entries equal, sox=0a13=b13y+1=11ory=10a23=b23quickExam plesRow Matrix , Column Matrix , and Square MatrixA Matrix with a single row is called a row Matrix ,or row Matrix with a sin-gle column is called a column matrixor column Matrix with the same num-ber of rows as columns is called a square 1 5matrix C=[3 401 11]is a row 4 1matrix D= 210 18 is a column 3 3matrix E= 1 20014 4321 is a square we go in Example 1 that the Matrix equation 79x0 1y+1 = 7900 111 is really six equations in one: 7=7, 9=9, x=0, 0=0, 1= 1.

5 And y+1=11. Weused only the two that were interesting. + 7/17/06 4:24 PM Page 175 Matrix Addition and SubtractionThe first Matrix operations we discuss are Matrix addition and subtraction. The rules forthese operations are 3 Matrix Algebra and ApplicationsquickExamplesMatrix Addition and Subtraction Two matrices can be added (or subtracted) if and only if they have the same add (or subtract) two matrices of the same dimensions, we add (or subtract) the cor-responding entries. More formally, if Aand Bare m nmatrices, then A+BandA Bare the m nmatrices whose entries are given by:(A+B)ij=Aij+Bijijth entry of the sum =sum of the ijth entries(A B)ij=Aij Bijijth entry of the difference =difference of the ijth entriesVisualizing Matrix AdditionExample2 SalesThe A-Plus auto parts store chain has two outlets, one in Vancouver and one in other things, it sells wiper blades, windshield cleaning fluid, and floor mats. Themonthly sales of these items at the two stores for two months are given in the follow-ing tables:VancouverQuebecWiper Blades2015 Cleaning Fluid (bottles)1012 Floor Mats84 January SalesVancouverQuebecWiper Blades2312 Cleaning Fluid (bottles)812 Floor Mats45 February Sales 2 310 + 11 21 = 3 2 11 1.

6 2 310 13 + 9 5013 13 = 11 8113 26 Corresponding entries added2. 2 310 13 9 5013 13 = 721 1300 Corresponding entries 7/17/06 4:24 PM Page 176 Scalar MultiplicationA Matrix Acan be added to itself because the expression A+Ais the sum of two ma-trices that have the same dimensions. When we compute A+A, we end up doublingevery entry in A. So we can think of the expression 2 Aas telling us to multiply everyelement in A by2. In general, to multiply a Matrix by a number, multiply every entry in the Matrix bythat number. For example,6 52 310 156 = 15 1860 65 It is traditional when talking about matrices to call individual numbers thisreason, we call the operation of multiplying a Matrix by a number scalar Matrix Addition and Scalar Multiplication177 Use Matrix arithmetic to calculate the change in sales of each product in each store fromJanuary to tables suggest two matrices:J= 20 1510 1284 andF= 23 1281245 To compute the change in sales of each product for both stores, we want to subtract cor-responding entries in these two matrices.

7 In other words, we want to compute the differ-ence of the two matrices:F J= 23 1281245 20 1510 1284 = 3 3 20 41 Thus, the change in sales of each product is the following:VancouverQuebecWiper Blades3 3 Cleaning Fluid (bottles) 20 Floor Mats 41using TechnologySee the Technology Guides atthe end of the chapter to seehow to add and subtract matricesusing a TI-83/84or , use the MatrixAlgebra Tool atChapter 3 Tools Matrix Algebra ToolThere, first enter the two matri-ces you wish to add or subtract(subtract, in this case) as shown:J=[20, 15, 1012, 8,4]F=[23, 12, 812, 4,5]To compute their difference, typeF-Jin the formula box and press Compute. (You can enter multi-ple formulas separated by com-mas in the formula box. For in-stance, F+J, F-Jwill computeboth the sum and difference.)Example3 SalesThe revenue generated by sales in the Vancouver and Quebec branches of the A-Plusauto parts store (see Example 2) was as follows:VancouverQuebecWiper Sales in Canadian 7/17/06 4:24 PM Page 177 Formally, scalar multiplication is defined as follows:178 Chapter 3 Matrix Algebra and ApplicationsIf the Canadian dollar was worth $ at the time, compute the revenue in need to multiply each revenue figure by Let Abe the Matrix of rev-enue figures in Canadian dollars:A= The revenue figures in dollars are then given by the scalar = In other words, in dollars, $91 worth of wiper blades was sold in Vancouver, $ of wiper blades was sold in Quebec, and so TechnologySee the Technology Guides at theend of the chapter to see how tocompute scalar multiples using aTI-83/84or ,go to the online Matrix AlgebraTool atChapter 3 Tools Matrix Algebra ToolThere, enter the January sales Dollars.

8 A=[140, 10530, 3696, 48]Then type *Ain the for-mula box and press Compute. Scalar MultiplicationIf Ais an m nmatrix and cis a real number, then cAis the m nmatrix obtained bymultiplying all the entries of Aby c. (We usually use lowercase letters c, d, e, ..todenote scalars.) Thus, the ijthentry of cAis given by(cA)ij=c(Aij)Example4 Combining OperationsLet A= 2 1035 3 ,B= 13 15 60 , and C= xywzt+13 Evaluate the following: 4A,xB, and A+ , we find 4 Aby multiplying each entry of Aby 4:4A=4 2 1035 3 = 8 401220 12 Similarly, we find xBby multiplying each entry of Bby x:xB=x 13 15 60 = x3x x5x 6x0 We get A+3 Cin two steps as follows:A+3C= 2 1035 3 +3 xywzt+13 In words, this rule is: To get the ijth entry of cA, multiply the ijth entry of Aby 7/17/06 4:24 PM Page 178 Addition and scalar multiplication of matrices have nice properties, reminiscent ofthe properties of addition and multiplication of real numbers. Before we state them, weneed to introduce some more Ais any Matrix , then Ais the Matrix ( 1)A.

9 In other words, Ais Amultipliedby the scalar 1. This amounts to changing the signs of all the entries in A. For example, 4 20610 6 = 420 6 10 6 For any two matrices Aand B, A Bis the same as A+( B).(Why?)Also, a zero matrixis a Matrix all of whose entries are zero. Thus, for example, the2 3zero Matrix isO= 000000 Now we state the most important properties of the operations that we have beentalking Matrix Addition and Scalar Multiplication179= 2 1035 3 + 3x3y3w3z3t+39 = 2+3x 1+3y3w3+3z3t+86 Properties of Matrix Addition and Scalar MultiplicationIf A, B, and Care any m nmatrices and if Ois the zero m nmatrix, then the fol-lowing hold:A+(B+C)=(A+B)+CAssociative lawA+B=B+ACommutative lawA+O=O+A=AAdditive identity lawA+( A)=O=( A)+AAdditive inverse lawc(A+B)=cA+cBDistributive law(c+d)A=cA+dADistributive law1A=AScalar unit0A=OScalar zeroThese properties would be obvious if we were talking about addition and multiplicationof numbers,but here we are talking about addition and multiplication of matrices.

10 Weare using + to mean something new: Matrix addition. There is no reason why matrixaddition has to obey allthe properties of addition of numbers. It happens that it doesobey many of them, which is why it is convenient to call it additionin the first means that we can manipulate equations involving matrices in much the same waythat we manipulate equations involving numbers. One word of caution: We haven t yetdiscussed how to multiply matrices, and it probably isn t what you think. It will turn outthat multiplication of matrices does notobey all the same properties as multiplication 7/17/06 4:24 PM Page 179 TranspositionWe mention one more operation on matrices:180 Chapter 3 Matrix Algebra and ApplicationsquickExamplesTranspositionIf Ais an m nmatrix, then its transposeis the n mmatrix obtained by writing itsrows as columns, so that the ith row of the original Matrix becomes the ith column of thetranspose. We denote the transpose of the Matrix Aby TranspositionProperties of TranspositionIf Aand Bare m nmatrices, then the following hold: (A+B)T=AT+BT(cA)T=c(AT)(AT)T=ATo see why the laws of transposition are true, let us consider the first one: (A+B)T=AT+BT.