Chapter 9  Matrices and Transformations
                   9MATRICES AND
                          TRANSFORMATIONS
                   Objectives
                   After studying this chapter you should
                   •   be able to handle matrix (and vector) algebra with confidence,
                       and understand the differences between this and scalar algebra;
                   •   be able to determine inverses of 2×2 matrices, recognising
                                                          
                       the conditions under which they do, or do not, exist;
                   •   be able to express plane transformations in algebraic and
                       matrix form;
                   •   be able to recognise and use the standard matrix form for less
                       straightforward transformations;
                   •   be able to use the properties of invariancy to help describe
                       transformations;
                   •   appreciate the composition of simple transformations;
                   •   be able to derive the eigenvalues and eigenvectors of a given
                       2×2 matrix, and interpret their significance in relation to an
                         
                       associated plane transformation.
                   9.0        Introduction
                   A matrix is a rectangular array of numbers.  Each entry in the
                   matrix is called an element.  Matrices are classified by the
                   number of rows and the number of columns that they have; a
                   matrix A with m rows and n columns is an m×n (said 'm by n')
                                                                 
                   matrix, and this is called the order of A.
                   Example
                   Given
                                      142
                                               
                                A =               ,
                                               
                                      3−10
                                               
                   then A has order 2×3 (rows first, columns second.)  The elements
                                       
                   of A can be denoted by a , being the element in the ith row and
                                             ij 
                   jth column of A.  In the above case, a   =1, a  =0, etc.
                                                         11      23
                                                          
                                                                                                                     235
                            Chapter 9  Matrices and Transformations
                            Addition and subtraction of matrices is defined only for matrices of
                            equal order; the sum (difference) of matrices A and B is the matrix
                            obtained by adding (subtracting) the elements in corresponding
                            positions of A and B.
                            Thus
                                                      142 −123
                                                                                                     
                                              A =                       and  B =
                                                                                                     
                                                      3−10                              43−3
                                                                                                     
                                                           065                                      22−1
                                     ⇒                                                         
                                              A +B=                           and  A −B =                              .
                                                                                               
                                                           72−3                                   −1−43
                                                                                               
                            However, if
                                                      23
                                                     
                                              C =               ,
                                                     
                                                      14
                                                     
                            then C can neither be added to nor subtracted from either of A or B.
                            If you think of matrices as stores of information, then the addition
                            (or subtraction) of corresponding elements makes sense.
                            Example
                            A milkman delivers three varieties of milk Pasteurised (PA), Semi-
                            skimmed (SS) and Skimmed (SK)) to four houses (E, F, G and H)
                            over a two-week period.  The number of pints of each type of milk
                            delivered to each house in week 1 is given in matrix M, while N
                            records similar information for week 2.
                                                   PA   SS   SK                          PA   SS   SK
                                               E 843                               E     478
                                                                                                    
                                                                                                    
                                               F 12 0 3                           F 10 0 5
                                                                 =M                                    =N
                                                    276                                 087
                                              G                                  G                  
                                                                                      
                                              H 690                                H 8100
                                                                                                    
                                                                                         
                            Then
                                                            12 11 11
                                                                           
                                                           22      0     8
                                              M +N =                       
                                                            21513
                                                                           
                                                            14 19 0
                                                           
                            records the total numbers of pints of each type of milk delivered to
                            each of the houses over the fortnight,
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                                                                                 Chapter 9  Matrices and Transformations
                  and
                                  −435
                                  
                                  
                                  −202
                         N −M =           
                                  
                                  −211
                                  
                                   210
                                  
                  records the increase in delivery for each type of milk for each of
                  the houses in the second week.
                  Suppose now that we consider the 3×2 matrix, P, giving the prices
                                                      
                  of each type of milk, in pence, as charged by two dairy companies:
                                           12
                                     PA 35 36
                                     SS 32 30=P
                                                
                                     SK 27 27
                                                
                                           
                  What are the possible weekly milk costs to each of the four
                  households?
                  Define the cost matrix as
                                          12
                                         cc
                                      E 
                                         11  12 
                                        c    c
                                      F
                                         21   22 = C
                                        c    c 
                                      G
                                         31   32
                                         c    c
                                      H
                                          41  42
                  Now c  is the cost to household E if company 1 delivers the milk
                         11
                         (in the week for which the matrix M records the deliveries)
                  and so
                               c  =8×35+4×32+3×27
                                11
                                  =489p
                  Essentially this is the first row        of M 'times' the first
                                                 843
                                                 []
                                                   
                          35
                  column 32 of P.
                             
                          27
                             
                            
                  Similarly, for example, c  can be thought of as the 'product' of the
                                           32
                                                                          36
                  third row of M,  276                                    30
                                            , with the second column of P,     , so
                                  []
                                                                            
                                                                          27
                                                                            
                                                                            
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                            Chapter 9  Matrices and Transformations
                            that                c   =2×36+7×30+6×27
                                                 32
                                                    =444p
                            This is the cost to household G if they get company 2 to deliver
                            their milk.
                            Matrix multiplication is defined in this way.  You will see that
                            multiplication of matrices X and Y is only possible if
                                                                             =  the number of rows of Y
                                      the number of columns X 
                            Then, if X is an  a×b  matrix and B a  c×d  matrix, the
                                                   ()                               ()
                                                                                      
                            product matrix XY exists if and only if b = c and XY is then an
                                                                                       
                             a×d  matrix.  Thus, for P = XY ,
                            ()                                     
                              
                                              P= p ,
                                                    ()
                                                       ij
                                                
                            where the entry p  is the scalar product of the ith row of X
                                                     ij
                                                     
                            (taken as a row vector) with the jth column of Y (taken as a
                            column vector).
                            Example
                            Find AB when
                                                                                   25
                                                      142  
                                                                      
                                              A =                      ,  B =      20
                                                                      
                                                      3−10
                                                                      
                                                                                 −13
                                                                                 
                            Solution
                            A is a 2×3 matrix, B is a 3×2 matrix.  Since the number of
                                                                    
                            columns of A   = the number of rows of B, the product matrix AB
                            exists, and has order 2×2.
                                                            
                                                            p        p 
                                              P=AB = 11               12
                                                            p21     p22
                                                
                                                                       2
                                               p = 142 .  2=2+8−2=8
                                                11   [] , etc
                                                                           
                                                                           
                                                                        −1
                                                                      
                            giving
                                                     811
                                              P= 
                                                     
                                                     415
                                                     
                            238