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# Note for MATH 3321 at UH

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Date Created: 02/06/15

55 Matrices and Vectors In the preceding section we introduced the concept of a matrix and we used matrices augmented matrices as a shorthand notation for systems of linear equations In this and the following sections we will develop this concept furtheri Recall that a matrix is a rectangular array of objects arranged in rows and columns The objects are called the entries of the matrix A matrix with m rows and n columns is called an m X n matrix The expression m X n is called the size of the matrix and the numbers m and n are its dimensions A matrix in which the number of rows equals the number of columns m n is called a square matrix of order n Until further notice we will be considering matrices whose entries are real numbers We will use capital letters to denote matrices and lower case letters to denote its entries If A is an m X n matrix then aij denotes the element in the ith row and j h column of A an a12 dis aln a21 a22 a23 a2n A a31 a32 ass 39 39 39 an aml am2 ams 39 39 39 amn We will also use the notation A aij to represent this display There are two important special cases that we need to de ne A 1 X n matrix a1 a2 Hi an also written as a1 a2 Hi an since there is only one row we use only one subscript is called an n dimensional row vector An m X 1 matrix a1 a2 am is called an m dimensional column vector The entries of a row or column vector are often called the components of the vector On occasion we7ll regard an m X n matrix as a set of m row vectors or as a set of n column vectors To distinguish vectors from matrices in general well use lower case boldface letters to denote vectors Thus we7ll write a1 a2 ua1a2iuan and v am Arithmetic of Matrices Before we can de ne arithmetic operations for matrices addition subtraction etc we have to de ne what it means for two matrices to be equal 187 DEFINITION Equality for Matrices Let A aij be an m X n matrix and let B bij be a p X 4 matrix Then A B if and only if 1 m p and n L that is A and B must have exactly the same size and 2 aij bij for all i and j In short A B if and only if A and B are identical Example 1 a b 3 7 71 I ifandonlyif a7b7lc4z3i I 2 c 0 2 4 0 Matrix Addition Let A aij and B blj be matrices of the same size say m X n Then A B is the m X n matrix C cij where cij aij by for all i and j That is you add two matrices of the same size simply by adding their corresponding entries A B aij Iij Addition of matrices is not de ned for matrices of di ererit sizes a b I y a z b y E l 2 xampe a lt0 dgtltz w 62 dw 2 4 73 74 0 6 72 4 3 b lt 2 5 0 gt lt 71 2 0 gt lt 1 7 0 gt 1 3 2 4 73 c 5 73 is not de ned I 2 5 0 0 6 Since we add two matrices simply by adding their corresponding entries it follows from the properties of the real numbers that matrix addition is commutative and associative That is if A B and C are matrices of the same size then A B B A Commutative A B C A B C Associative A matrix with all entries equal to 0 is called a zero matrix For example 0 0 0 0 0 0 0 0 and 0 0 0 0 are zero matricesi Often the symbol 0 will be used to denote the zero matrix of arbitrary size The zero matrix acts like the number zero in the sense that if A is an m X n matrix and 0 is the m X n zero matrix then A00AAi 188 A zero matrix is an additive identity The negative of a matrix A denoted by 7A is the matrix Whose entries are the negatives of the entries of Al For example if a b c 7a 7b 7c A th 7 A def en lt7d 7e 7f Subtraction Let A aij and B bij be matrices of the same size say m X n Then A 7 B A 7B To put this simply A 7 B is the m X n matrix C clj Where cij aij 7 by for all i and j you subtract two matrices of the same size by subtracting their corresponding entries A 7 B aij 7 Iij Note that A7 A 0i Subtraction of matrices is not de ned for matrices of di erent sizesi 2473 740676479 I 250 71207330 Multiplication of a Matrix by a Number Example 3 The product of a number k and a matrix A denoted by kA is the matrix formed by multiplying each element of A by the number kl That is kA kaij This product is also called multiplication by a scalar Example 4 71 4 76 3 712 73 l 5 72 73 715 6 i I 4 0 3 712 0 79 Matrix Multiplication We now want to de ne matrix multiplication While addition and subtraction of matrices is a natural extension of addition and subtraction of real numbers matrix multiplication is a much different sort of product We7ll start by de ning the product of an ndimensional row vector and an ndimensional column vector in that speci c order 7 the row on the left and the column on the right The Product of a Row Vector and a Column Vector The product of a l X n row vector and an n X 1 column vector is the number given by a1 a2 a3 Hi an b3 a1b1 agbg agbg anbni 189 This product has several names including scalar product because the result is a number scalar dot product and inner product It is important to understand and remember that the product of a row vector and a column vector of the same dimension and in that orderl is a number The product of a row vector and a column vector of dz erent dimensions is not de ned Example 5 71 3 72 5 74 371727451 10 l 2 72 3 71 4 f3 722347173475 79 75 Matrix Multiplication If A aij is an m X p matrix and B blj is a p X 71 matrix then the matrix product of A and B in that order denoted AB is the m X n matrix C cij Where cij is the product of the ith row of A and the jth column of B an a12 quot39 a1 1111 39 39 39 blj 39 39 39 bln 1121 39 39 39 112139 39 39 39 1127 an L112 aw C Cij bpl 39 39 39 bpj 39 39 39 bzm am am amp Where cij ailblj aigbgj aipbm l NOTE Let A and B be given matrices The product AB in that order is de ned if and only if the number of columns of A equals the number of rows of B If the product AB is de ned then the size of the product is no of rows of Agtltnol of columns of B That is A B C mgtltp pgtltn mgtltn Alt j and BE 32 Since A is 2 X 3 and B is 3 X 2 we can calculate the product AB Which Will be a 2 X 2 matrix 3 0 AB 7 1 4 2 71 2 1347121 1042272 31 5 1 2 33171513012572 33 Example 6 Let 190 We can also calculate the product BA since B is 3 X 2 and A is 2 X 31 This product will be 3X31 3 0 3 12 6 BA 71 2 lt1 4 2 5 72 6 i I 172 315 75276 Example 6 illustrates a signi cant fact about matrix multiplication Matrix multiplication is not commutative AB BA in general While matrix addition and subtraction mimic addition and subtraction of real numbers matrix multiplication is distinctly different Example 7 a 1n Example 6 we calculated both AB and BA but they were not equal because the products were of different size Consider the matrices 12 7103 Clt34gt and Dlt5 72gt 9147 Here the product CD exists and equals 17 28 17 gt You should verify this On the other hand the product DC does not exist you cannot multiply D C 1 2X3 2X2 E 4 71 and 9 75 i 0 2 3 0 In this case EF and FE both exist and have the same size 2 X 21 But EF lt 33 720 gt and FE lt 36 719 gt verify b Consider the matrices 6 0 12 73 so EFa FEi I While matrix multiplication is not commutative it is associative Let A be an m X p matrix B a pgtlt 4 matrix and C a 4X n matrixi Then ABC ABC1 By indicating the dimensions of the products we can see that the left and right sides do have the same size namely m X n A BC and AB Ci quotIX pgtltn mgtltq 11X A straightforward but tedious manipulation of double subscripts and sums shows that the left and right sides are in fact equali There is another signi cant departure from the real number systemi If a and b are real numbers and ab 0 then either a 0 b 0 or they are both 01 Example 8 Consider the matrices A071andB351 02 00 191 Neither A nor B is the zero matrix yet as you can verify 1900 00 07 WW As we saw above zero matrices act like the number zero with respect to matrix addition Also A00AAi Are there matrices which act like the number 1 with respect to multiplication a 1 1 a a for any real number a Since matrix multiplication is not commutative we cannot expect to nd a matrix I such that AI IA A for an arbitrary m X n matrix Al However there are matrices which do act like the number 1 with respect to multiplication Example 9 Consider the 2 X 3 matrix OH HO CO H 27 39D 5 Let I2lt1j and I3 10 abc abc 12Alt01gtltd e fgtltd e f b 100 b A13lta C 010 a a I def 001 def Identity Matrices Let A be a square matrix of order n The entries from the upper left corner O O H and of A to the lower right corner that is the entries an agg agg i i ann form the main diagonal of A For each positive integer n gt 1 let In denote the square matrix of order n whose entries on the main diagonal are all 1 and all other entries are 0 The matrix In is called the n X n identity matrix In particular 10 10 0 I2lt01gt 13 010 I4 001 and so on OOOH OOHO OHOO HOOD If A is an m X n matrix then ImA A and ALL AA 192 In particular if A is a square matrix of order n then ALL InA A so In does act like the number 1 for the set of square matrices of order n Matrix addition multiplication of a matrix by a number and matrix multiplication are connected by the following properties In each case assume that the sums and products are de ned ll AB C AB AC This is called the left distributive law 2 A BC AC BC This is called the right distributive law 3 MAB kAB AkB Another way to look at systems of linear equations Now that we have de ned matrix multiplication we have another way to represent the system of m linear equations in n unknowns 1111 1212 1313 ainrn 1 1 2111 2212 2313 a2n1n 1 2 3111 3212 3313 asnrn 1 3 amiri am212 amsrs amnrn bm We can write this system as an L112 an am 11 b1 L121 L122 L123 a2n I2 b2 L131 L132 ass asn Is b3 am am am am In bm or more formally in the vectormatrix form Ax b l where A is the matrix of coef cients x is the column vector of unknowns and b is the column vector whose components are the numbers on the righthand sides of the equations This looks like77 the linear equation in one unknown ax b where a and b are real numbers By writing the system in this form we are tempted to divide77 both sides of equation 1 by A provided A is not zero77 Well take these ideas up in the next section The set of vectors with n components either row vectors or column vectors is said to be a vector space of dimension n and it is denoted by Rnl Thus R2 is the vector space of vectors 193 with two components R3 is the vector space of vectors with three components and so on The zy plane can be used to give a geometric representation of R2 3dimensional space gives a geometric representation of Rgi Suppose that A is an m X n matrixi If u is an n component vector an element in R then Au v is an m component vector an element of Rm Thus the matrix A can be viewed as a transfor mation a function a mapping that takes an element in R to an element in Rm A maps R into Rmi 1724 7103 lt41gt ltgt7 lt32gtEltgt and in general 1 72 4 L 7 a72b4c 71 0 3 7a30 c If we view the m X n matrix A as a transformation from R to Rm then the question of Example 10 The 2 X 3 matrix A lt gt maps R3 into R For example solving the equation Ax b can be stated as Find the set of vectors x in R which are transformed to the given vector b in Rmi As a nal remark an m X n matrix A regarded as a transformation is a linear transformation since Ax y AX Ay and Akx kAxi These two properties follow from the connections between matrix addition multiplication by a number and matrix multiplication given above Exercises 55 2 4 720 LLetA 71 3 B 42 Clt172gtDlt7 72gt 472 731 Compute the following if possible a 43 b 72B c CD d 41 e 2AB f 473 71 0 72 3 76 5 2Let4 2 B 3 72 0 Clt72 4 73gtD 3 0 4 2 3 1 4 4 Compute the following if possible 194 a AB b BA c CB d CA e DA f DB g AC h BQBB 0 71 71 0 2 0 5 0 iLetA 0 2 B 3 72 C 7 D i 2 5 H SL3 Compute the following if possible a 3A 7 2BC b AB c BA d CD 7 2D e AC 7 BD f AD 2DC Let A be a 3 X 5 matrix B a 5 X 2 matrix C a 3 X 4 matrix D a 4 X 2 matrix and E a 4 X 5 matrix Determine which of the following matrix expressions exist and give the sizes of the resulting matrix when they do exist a AB b EB c AC d AB CD e 2EB 7 3D f CD 7 CEB g EB DA 0 73 5 71 72 3 1A 2 4 71 and B 5 4 73 iLet CAB and DBA1 1 0 2 0 2 5 Determine the following elements of C and D without calculating the entire matrices a 632 b 013 C d21 d d2 2 3 0 1 3 1A 4 75 and B 1 4 3 iLet CAB and DBA1 71 3 Determine the following elements of C and D without calculating the entire matrices a 021 b 033 c dlg d du iLetA 1 3 B 12 3 andC 2 3 5 iPutDAB201 2 4 5073 7 10 Determine the following elements of D without calculating the entire matricesi a dgg b dlg c d23 1 73 1 1 2 73 2 0 72 i Let A 72 4 0 B 1 1 73 and C 4 5 71 1 Put 3 71 74 71 3 2 1 0 72 D 2B 7 301 Determine the following elements of D without calculating the entire matricesi a dll b d23 c d3 1 Let A be a matrix whose second row is all zeros Let B be a matrix such that the product exists AB exists Prove that the second row of AB is all zeros 195 10 11 12 13 14 15 Let B be a matrix Whose third column is all zeros Let A be a matrix such that the product exists AB exists Prove that the third column of AB is all zeros gtvBlt gtvolt gt7Dltgt Calculate if possible b AC and CA c AD and DA a AB and BA This illustrates all the possibilities When order is reversed in matrix multiplication 21 LetAlt gtBlt gto 34 712 Calculate ABC and ABCi This illustrates the associative property of matrix multipli 1 0 cationi Let A B 0 72 C i 0 1 3 5 Calculate ABC and ABCi This illustrates the associative property of matrix multipli cation 12 710 054 213 23 61 12 713 240 7213 723 30 72 74 Let A be a 4 X 2 matrix B a2 X 6 matrix C a 3 X 4 matrix D a6 X 3 matrix Determine Which of the following matrix expressions exist and give the sizes of the resulting matrix When they do exist a ABC d DCAB b ABD e AZBDC c CAB Let A be a 3 X 2 matrix B a 2 X 1 matrix C a 1 X 3 matrix D a 3 X 1 matrix and E a 3 X 3 matrix Determine Which of the following matrix expressions exist and give the sizes of the resulting matrix When they do exist a ABC d DAB AB b ABEAB e BCDCBC c DC 7 EAC 196

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