Matrices and Systems of Linear Equations

Transcription

1 Chapter Matrices and Systems of Linear Equations In Chapter we discuss how to solve a system of linear equations. If there are not too many equations or unknowns our task is not very difficult; what we learned in high school will suffice. Fortunately our world is fairly complicated. We quite often meet problems that can be reduced to solving a system of several hundred or more equations with an equally large number of unknowns. Thus our method of solution should simplify the bookkeeping involved and be as direct and straightforward as possible. This simplification process leads in a natural manner to objects called matrices and various rules for manipulating them. A great amount of time and effort will be spent on matrices, but we always need to keep in mind that we are discussing systems of linear equations.. Systems of Linear Equations Let s look at a simple example of a system of linear equations: 2x +3x 2 = 6 x +4x 2 = 8 (.) The problem is to determine all possible pairs of numbers, which we denote by x and x 2, such that these numbers will satisfy both equations in (.). The trick (technique) in solving (.) is to first remove one of the unknowns from one of the equations. To accomplish this, multiply the second equation by 2 and then add the resulting equation to the first. This gives us the following system: x 2 = 22 x +4x 2 = 8 (.2) This system of equations is easy to solve. Clearly x 2 = 22 = 2, and replacing x 2 with 2 in the second equation we have x +4(2) = 8 or x = 0. Hence,

2 2 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS x = 0. We now check that we have actually found a solution to our original system (.). That is, we replace x by 0 and x 2 by 2 and verify that both equations are satisfied. Let s go back to the beginning and try to reduce some of the work involved. We started with (.), but now instead of writing the unknowns every time let s just write the coefficients. This leads us to List coefficients 2x +3x 2 = 6 x +4x 2 = : 6 4 : 8 2R 2 +R 0 : 22 4 : 8 R R 2 4 : 8 0 : 22 R 4 : 8 R 2 0 : 2 (.3) The notation is almost self-explanatory. For example, 2R 2 +R means multiply the second row (equation) by 2 and then add it to the first row, which gives us a new first row (equation). R R 2 means interchange the first and second rows. R and R 2 mean multiply the first row by and the second row by. The colons are used to distinguish the right-hand side of equations (.3) from the coefficients of the unknowns. Later we will not bother with the colons. The next step in our procedure is to realize that the final array in (.3) represents the equations x 4x 2 = 8 x 2 = 2 (.4) From (.4) we again get the solution x 2 = 2, x = 8+8 = 0. The form of this final array should be noted; namely, there are ones in the main diagonal and zeros below it, which means that we can easily solve for x 2 and then x. Let s look at one more specific example before discussing linear systems in general. Example. 2x +3x 2 x 3 = 6 4x + x 3 = 7 x 2 2x 3 = (.5) Remember that we re going to delete the variables and manipulate the rows in order to arrive at the following form if possible.. : b 0. : b : b 3 (.6)

3 .. SYSTEMS OF LINEAR EQUATIONS 3 From (.5) we have 2R +R 2 R 2 R 3 6R 2 +R 3 2 R R : : : 2 3 : : : 2 3 : : 0 6 : : : 0 0 : R 3 2 : : 0 0 : 3 (.7) With this last array we associate the following system: x x 2 2 x 3 = 3 x 2 2x 3 = (.8) x 3 = 3 Sincethe leadingcoefficientsareone, wecaneasilysolvethis systemforx 3 = 3, x 2 = +2( 3 ) = 37, and x = 3+ 2 x x 2 = 3+( 2 )(3 ) (3 2 )(37 ) = 6. The ease with which (.8) is solved demonstrates why the form in (.6) is desirable. Question: The numbers x,x 2, and x 3 satisfy (.8). Do they also satisfy (.5)? They had better! The fact that they do can be demonstrated by either putting these numbers into equations (.5) or observing that the row operations, which were performed, are all reversible. For example, the reverse (inverse) of 6R 2 + R 3 would be 6R 2 + R 3. Starting with the final array of (.7), if we execute the sequence of operations R 3, 2R, 6R 2 +R 3,R 3 R 2, 2R +R 2, we will get the original array. Since (.8) may be algebraically manipulated to get (.5), every solution of (.8) will also satisfy (.5), and conversely every solution of (.5) will also be a solution of (.8). We next discuss the form of a general system of m linear equations in n

4 4 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS unknowns: a x +a 2 x 2 + +a n x n = b a 2 x 2 +a 22 x 2 + +a 2n x n = b 2... a m x +a m2 x 2 + +a mn x n = b m (.9) Thus symbols x,x 2,...,x n represent the unknowns that we are trying to find. The a ij, i m, j n, are called the coefficients of the system. It is assumed that the coefficients a ij and the terms b j, j m, are all known. A few words about the subscripts are in order. The first subscript, i, of a ij refers to the equation in which this term appears, while the second subscript, j, tells us which of the n unknowns a ij multiplies. Thus, if a 23 = 6 we know that the variable x 3 in the second equation is multiplied by 6. To make sure this convention is understood, let s go back to (.5): 2x +3x 2 x 3 = 6 4x + x 3 = 7 x 2 2x 3 = For the above system of equations we would have a = 2 a 2 = 3 a 3 = b = 6 a 2 = 4 a 22 = 0 a 23 = b 2 = 7 a 3 = 0 a 32 = a 33 = 2 b 3 = (.5) One more bit of terminology. The procedure we ve been using is referred to as Gaussian elimination. We summarize it below.. List the coefficients only. 2. Use row operations to obtain zeros below the main diagonal. 3. Solve the new system. The phrase main diagonal in 2 above refers to those a ij for which i = j. Example 2. This last matrix implies that x +x 2 x 3 = x + x 3 = 0 [ ] : 0 : 0 R +R : [ ] : R 2 [ ] : 0 2 : R 2 +R 0 2 : [ ] 0 : 0 x = x 3 x 2 = +2x 3

5 .. SYSTEMS OF LINEAR EQUATIONS 5 Here we did not get numbers for an answer, but equations relating two of the unknowns to the third. This means that we have an infinite number of solutions, a different one for each choice of x 3. For example, if we set x 3 = 0, then x = 0 and x 2 = is a solution. Similarly the triple x 3 =, x 2 = 3, x = is also a solution. Example 3. R +R 3 R 2 +R 3 x + x 2 = x 2 +3x 3 = 4 x +2x 2 +3x 3 = 6 0 : 0 3 : : 6 0 : 0 3 : : 5 0 : 0 3 : : Now what? This last array represents the system of equations: x + x 2 = x 2 +3x 3 = 4 0x +0x 2 +0x 3 = Clearly the third equation has no solution, which means that our original system can have no solution. These examples indicate that for an arbitrary linear system we may have one of three possibilities:. No solution: Example 3 2. Exactly one solution: Example 3. An infinite number of solutions: Example 2 It is not clear that these are the only possibilities. For example, is there a system with exactly two solutions or some other finite number? We shall see later that this cannot happen, and that the above three cases are the only possible ones. Before concluding this first section we discuss in the next example the geometry of systems of equations, at least in two unknowns.

6 6 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS Example 4. Sketch the solutions of each of the following systems of equations: a. 2x +x 2 = 4x +2x 2 = 2 x 2 (0,) 4x + 2x 2 = 2 2x + x 2 = ( 2,0) x The locus of points that satisfy this system is a straight line as shown above. b. 2x +x 2 = x x 2 = 0 x 2 x x 2 = 0 ( 3, 3 ) x 2x + x 2 = The locus of points that satisfy both equations is all points common to both straight lines. In this case there is only one such point, namely, x = 3 = x 2. c. 2x +x 2 = 2x +x 2 = 4 x 2 2x +x 2 = 4 2x +x 2 = x

7 .. SYSTEMS OF LINEAR EQUATIONS 7 Each equation describes a straight line, and these lines are parallel. Since they do not intersect, the system of equations has no solution. Problem Set.. Find all solutions, if any, of the following systems of linear equations: a. 3x 6x 2 = 0 2x +4x 2 = 0 b. 3x 6x 2 = 3 2x +4x 2 = 2 2. Solve the following systems of linear equations: a. 2x +3x 2 = 7 6x + x 2 = b. 2x +3x 2 = b 6x + x 2 = b 2 3. Find all solutions, if any, of the following systems of linear equations: a. 2x +x 2 x 3 = x +x 2 +x 3 = 2 b. 2x +x 2 x 3 = x +x 2 +x 3 = 2 x x 2 = 2 4. Find all numbers k such that the following system of equations has a solution and then solve that system. 2x + x 2 x 3 = x + x 2 +x 3 = 2 3x +2x 2 = k 5. Find all solutions of the following system of linear equations: 2x +3x 2 x 3 + x 4 = 2 x + x 2 + x 3 2x 4 = 6 x + 2x 3 +4x 4 = 3 2x 2 + x 3 x 4 = 5 6. Find all solutions of the following system: x x 3 + x 4 = 0 4x + x 3 +2x 4 = 0 x +2x 2 x 3 x 4 = 0 2x 2 +5x 3 4x 4 = 0

8 8 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS 7. Matt s, Dave s, and Abe s ages are not known but are related as follows: The sum of Dave s and Abe s ages is 3 more than Matt s. Matt s age plus Abe s age is 9 more than Dave s. If the sum of their ages is 35, how old is Abe? 8. Granny s favorite recipe for pecan pie calls for cup of pecans, 3 cups of sugar, and 3 eggs. Her recipe for one batch of pecan puffs calls for cup of sugar, cup of pecans, and 2 eggs. If she has 2 pounds of pecans, 6 pounds of sugar, and a henhouse next door (i.e., as many eggs as she needs), how many pecans pies and batches of pecan puffs can she make while using up all her supplies? ( pound = 2 cups) 9. Two cars are traveling toward each other. The first is moving at a speed of 45 mph and the second at 65 mph. If, at the start, they are 35 miles apart, how long before they meet and how far will each car have traveled? 0. Find all solutions of the following system: 2x + x 3 +6x 4 = 0 x +3x 2 +4x 3 +2x 4 = 0 x 2 +3x 3 +4x 4 = x + x 2 x 3 x 4 =. Find all solutions of the following system of equations: 2x +2x 2 + x 4 = 0 x x 2 +2x 3 + x 4 = x + x 2 + 4x 4 = 2. If possible, solve the following systems of equations: a. 3x 2x 2 +3x 3 = 0 x +x 2 +x 4 = 0 x x 3 x 4 = 0 b. x x 2 = x +x 2 x 3 = 2 x 5x 2 +2x 3 = 3. Dabs cement plant makes mortar and cement. Each ton of mortar requires 2 ton of lime and 5 ton of sand. Each ton of cement requires 3 ton of lime and 3 4 ton of sand. If Dabs has 9 tons of lime and tons of sand available, how many tons of mortar and cement can the company make? 4. Graph the solution sets for each of the following systems of equations. Since there are two unknowns, the solution set is the set consisting of all pairs of numbers (x,x 2 ) that satisfy the equations a. x 2x 2 = 6 2x + x 2 = b. 3x +4x 2 =

9 .. SYSTEMS OF LINEAR EQUATIONS 9 5. Graph the solution sets of the following systems of equations. Here, the solution set consists of those triples of numbers (x,x 2,x 3 ) that satisfy the equations a. x +x 2 +x 3 = 0 b. x +x 2 +x 3 = 0 x +x 3 = 0 c. x +x 2 +x 3 = 0 x +x 3 = 0 x x 3 = 0 6. Find an equation for the straight line that passes through the following points: a. (,6),(,4) b. ( 2,6),(4,3) 7. Is there a straight line that passes through the following sets of points? a. (,6),(,4),(3,7) b. ( 4,2),(3,7),(,8) 8. Find an equation for the circle that passes through the three points( 4, 2), (3,7), (,8). 9. Find the solution set to the equations a. x +x 2 = 0 b. x +x 2 = c. Graph the solution sets of a and b. How are they related? 20. Suppose we have an infinite sequence of numbers x k, k =,2,..., that are related to each other by the equations x k+ = 2x k. a. Find a formula for x k in terms of x. For example, x 2 = 2x and x 3 = 4x. b. Suppose the x k are related by the equation x k+2 = x k +x k+. Thus x 3 = x +x 2 and x 4 = x 2 +x 3 = x +2x 2. Can you find a formula for x k (k > 3) in terms of x and x 2? 2. Suppose we have two sequences of numbers x k and y k for k =,2,..., and that the sequences are related by the equations x k+ = x k +y k y k+ = x k y k Find a formula for x k and y k in terms of x and y for all k if you can. If you cannot discover such a formula, compute x 3 and y 3 in terms of x and y.

10 0 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS.2 Matrices In the preceding section the first step taken in solving a system of equations was to remove the variables, which left us with a rectangular array of numbers. These arrays are given a special name. Definition.. An m n matrix is a rectangular array of numbers arranged in m rows and n columns. In most of this book we use real numbers, but from time to time, as the inclination or need arises, complex number will appear. Example. 2 A = is a 2 2 matrix 3 4 [ B = is a 2 matrix 3] C = is a 3 4 matrix We will often denote an m n matrix A by writing A = [a ij ], i m, j n. The symbol a ij is used to indicate the entry in the ith row and jth column. Thus, counting from the top left corner of the matrix, the first subscript tells us which row and the second subscript which column the entry appears in. Example 2. 2 A = 3 4 [ b B = = 3] b 2 = 3 C = a =,a 2 = 2 a 2 = 3,a 22 = 4 c =,c 2 = 5,c 3 =,c 4 = 6 c 22 = 0,c 23 = 2,c 24 = 8 c 33 = 3,c 34 = There are various operations that we may perform involving matrices, and we will spend the next few pages defining and exemplifying them. Before doing so, however, we need to state clearly what it means to say that two matrices are equal. Let A and B be two matrices. We say A = B if they are of the same size and their corresponding entries are equal. For example, the matrices 2 0 and 2

11 .2. MATRICES are not equal since the entries in the second row and first column are not equal. The matrices 2 and 2 0 are not equal since the first is a 2 2 matrix and the second is a 2 matrix. A precise definition of matrix equality is: Definition.2. Let A = [a ij ], i m, j n. Let B = [b ij ], i p, j q, be two matrices. We say A = B if and only if m = p, n = q, and a ij = b ij for each i and j. One algebraic operation involving matrices is that of addition. Given two matrices of the same size, we form their sum by adding together their respective entries. Thus = and [ 2 6 ] + [ 4 ] = [ 2 0 ] A formal definition of matrix addition is: Definition.3. Let A = [a ij ], i m, j n. Let B = [b ij ], i m, j n. The sum S = A+B of these two m n matrices is the m n matrix whose i,j entry, s ij, equals a ij +b ij. Example a. + = [ ] b = There are also times when we wish to multiply a matrix by a number. To do this, just multiply every entry in the matrix by that number. For example, = Definition.4. Let A = [a ij ] be any m n matrix and c any number. Then ca is the m n matrix whose i,j entry is ca ij. Example a = b. =

12 2 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS As with real or complex numbers, these two operations satisfy certain laws, which we state in the following theorem: Theorem.. Let A = [a ij ], B = [b ij ], and C = [c ij ] be any m n matrices. Let a and b be any numbers. Then the following properties hold:. A+B = B +A 2. (A+B)+C = A+(B +C) 3. a(a+b) = aa+ab 4. (a+b)a = aa+ba 5. a(ba) = (ab)a Proof. The proofs of these statements are straightforward, and we shall go through the details of properties and 3 only. A+B = [a ij ]+[b ij ] = [a ij +b ij ] = [b ij +a ij ] = B +A 3. a(a+b) = a([a ij ]+[b ij ]) = a([a ij +b ij ]) = [a(a ij +b ij )] = [aa ij +ab ij ] = [aa ij ]+[ab ij ] = a[a ij ]+a[b ij ] = aa+ab The next example repeats the above computations using particular matrices and numbers. Example 5. ( ) = 3 = = =

13 .2. MATRICES 3 We see that our answer does not depend upon whether we first add and then multiply, or multiply and then add. There is a special m n matrix, called the zero matrix, which we denote by O mn (when the size is clear we will drop the subscripts). This matrix has all its entries equal to zero. Thus, [ O 2 = and O 0] 22 = 0 0 It is clear that the following is true: Theorem.2. Let A be any m n matrix. Then. A+O mn = A 2. 0A = O mn Our next task is to learn how to multiply matrices. The rule at first seems unusual, but after we relate it in the next section to systems of equations, it will be clear that we are not being unreasonable. The first thing to remember is that two arbitrary sized matrices A and B cannot be multiplied together to form AB. The number of columns in the first factor A must equal the number of rows in the second factor B. To calculate the i,j entry of AB, we multiply the entries of the ith row of A by the corresponding entries of the jth column of B, and then sum these products. For example, if [ 2 2 A = B = 4 3 6] then AB will be the 2 matrix C, where [ c (first row of A) times (first column of B) C = = c 2 (second row of A) times (first column of B) or [ 2 2 ] [ 2 ] [ ()( 2)+(2)(6) = ( 4)( 2)+(3)(6) The precise definition of matrix multiplication is: ] = 0 26 Definition.5. Let A = [a ij ] be an m n matrix. Let B = [b ij ] be an n p matrix. Then AB = C = [c ij ] is an m p matrix where ] c ij = a i b j +a i2 b 2j + +a in b nj n = a ik b kj k= Note the respective sizes of these matrices:

14 4 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS (m n)(n p) = m p The number of columns in A must equal the number of rows in B. The number of rows in AB is the same as the number of rows in A and the number of columns in AB is the same as the number of columns in B. Readers unfamiliar with summation notation should refer to Appendix B. Figure. illustrates this definition. where a i b. j b 2j... a in.. b ij = c ij.... b nj. c ij = a i b j +a i2 b 2j + +a in b nj Figure. Example 6. 0 a = b. = Note that we cannot multiply these two matrices in opposite order since the number of columns (three) in the second matrix does not equal the number of rows (two) in the first matrix. [ 0 0 ] 2 3 = 4 [ ] = [ Multiplying these matrices in the opposite order, we have c. = = Note that the products in b and c are not equal. Thus, even though AB and BA are both defined, they may not be equal. Matrix multiplication does satisfy some algebraic laws, which are stated in the following theorem: ]

15 .2. MATRICES 5 Theorem.3. Let A,B, and C denote matrices of appropriate size; that is, if we write AB, then A has the same number of columns as B has rows. Let a be any real number. Then the following formulas hold:. (A+B)C = AC +BC,A(B +C) = AB +AC 2. (AB)C = A(BC) 3. a(ab) = aa(b) = A(aB) Proof. Let A = [a ij ] be an m n matrix, B = [b ij ] be an m n matrix, and C = [c ij ] be an n p matrix. Then (A+B)C = [a ij +b ij ][c ij ] [ n ] = (a ik +b ik )c kj ) k= [ n = a ik c kj + k= ] n b ik c kj k= [ n [ n ] = a ik c kj ]+ b ik c kj k= = AC +BC The proof that A(B + C) = AB + AC is essentially the same as that for (A+B)C = AC +BC; so we omit it. Properties 2 and 3 may be proved in a similar manner, and the reader is asked to do so in the problems at the end of this section. We ve seen that there is a zero matrix O mn. There is also an identity matrix I n which plays the role of the number in matrix multiplication. The n n matrices, I n, are defined as follows: I = [] I 2 = 0 0 For an arbitrary natural number n we define k= I 3 = where δ ij = { if i = j 0 if i j I n = [δ ij ] That is, the symbol δ ij, which is called the Kronecker delta, is equal to if i equals j and zero if the subscripts are different.

16 6 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS Example 7. Let A = a b be any 2 2 matrix. Show that I c d 2 A = AI 2 = A. 0 a b a+0 b+0 a b I 2 A = = = = A 0 c d 0+c 0+d c d a b 0 a+0 0+b a b AI 2 = = = = A c d 0 c+0 0+d c d Theorem.4. Let A be an m n matrix. Let I n and I m denote the n n and m m identity matrices, respectively. Then I m A = AI n = A Proof. Let A = [a ij ]. Then [ m ] I m A = [δ ij ][a ij ] = δ ik a kj = [a ij ] = A Since δ ij = 0 unless k = i, the sum of the m terms δ ik a kj has only one nonzero summand, the one for which k = i. Similarly AI n = A. Note that if A is a square n n matrix, I n A = AI n = A. By this time the reader should have the feeling that if a rule is true for numbers it is also true for matrices. This feeling should be maintained with two important exceptions:. AB BA This does not mean that AB always differs from BA but rather that equality need not hold. 2. AB = AC does not imply that B = C The reader will find examples of this in the problems at the end of this section. There is one other operation that we need to discuss. It s called taking the transpose of a matrix. The operation is very simple to perform. We merely interchange rows and columns; that is, the ith row of A become the ith column of A T (A transpose). For example, if 0 A = k= then A T = Definition.6. Let A = [a ij ] be an m nmatrix. The transposeof A, denoted by A T = [a T ij ], is an n m matrix where a T ij = a ji

17 .2. MATRICES 7 In the future rather than write 2 4 we will write [ 2 4 ] T. Example 8.. T 2 = [ 2 3 ] T = T 6 = The following theorem, whose proof we omit, is stated for future reference. Theorem.5. Let A and B be two matrices. Then Example 9. Let A = (AB) T = B T A T 2 and B = 0 3 two matrices. () T T (AB) T = = = B T A T = = Verify Theorem.5 for these Definition.7. A square matrix A is said to be symmetric if A = A T. Thus a 2 2 matrix a b A = c d is symmetric if and only if c = b. Problem Set Compute the fol Let A = and let B = 0 2 lowing matrices:

18 8 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS a. 3A b. A B c. 2A+3B 2. Perform the indicated operations. 6 7 a b c Let A = 2. Let B = Compute the following: 0 3 a. AB b. BA c. 2A B Let A =. Let B = 2 4. Compute if possible the following matrix products: AB, BA, A 2 = AA, and B 2 = BB. 5. Compute the following products: a. b Let A =. Are there any numbers x 2 and x 2 such that if B = x 0, then AB = BA? x 2 7. For each of the matrices below compute A 2 and A 3. Note A 2 = AA and A 3 = AA 2 = A 2 A. 0 a b a. b. c c d k +k 8. Find all numbers k such that the matrix K = satisfies the k k equation K 2 = I 2. a b 0 9. If A =, show that A c d 2 = is not possible for any choice of 0 0 a,b,c, or d.

19 .2. MATRICES 9 d 0 0. Let D =, where d is an arbitrary number. Show that DA = AD 0 d for any 2 2 matrix A. Conversely show that if D is a matrix for which AD = DA for every 2 2 matrix A, then D has the above form for some number d. Such matrices are called scalar matrices Let A =. Let B =. Do there exist matrices X or Y 4 0 such that the equations AX = B or YA = B have solutions? 2. Let A = and let B = Compute the following matrices: a. A 2,B 2,AB,BA b. A T,B T c. A 3,B 3 d. A 4,B 4 3. Let A be an arbitrary n n matrix. Define A 0 = I n, A = A, A 2 = AA, and A n+ = AA n, for n =,2,3,.... Show that if p and q are arbitrary nonnegative integers then A p A q = A p+q. Thus, we may conclude that A p A q = A q A p. (Hint: Use induction.) 4. Let A =, B = , C =, D = Calculate, when defined, each of the following matrices: a. 2A+B,2C D,B +D b. AB,BA c. A 2,B 2 d. A T,B T Let A =, B =. Show (AB) T = B T A T Let A =, X = [ 2 ]. Show that (XA 4 5 T ) T = AX T. 7. Prove properties 2 and 3 of Theorem Let A =, B =. Verify that A [ 7 2 ] T is the first column of AB and that A [ 5 ] T is the second column of AB. 9. Let A and B be the matrices in problem 8. Verify that [ 6 ] B is the first row of AB and that [ 4 3 ] B is the second row of AB. 20. Show that

20 20 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS a. 2 x 2 = x 6 7 x 2 6 b. [ x x 2 ] [ [ +x 2 7 ] = x 2 +x Let A be any 3 3 matrix whose last row has only zeros. Let B be any 3 3 matrix. Show that the third row of AB has only zeros in it. 22. Generalize problem Which of the following matrices is symmetric? a. b c Let A and B be two m n matrices. Show that (A+B) T = A T +B T. 25. Show that (A T ) T = A, for any matrix A. 26. Verify Theorem.5 for the matrices in problem Prove Theorem.5 for A, any 2 2 matrix, and B, any 2 3 matrix. 28. Show that if A and B are symmetric matrices of the same size, then (AB) T = BA. Find two symmetric 2 2 matrices A and B such that AB is symmetric; then find two more 2 2 symmetric matrices such that their product is not symmetric. 29. Show that if A and B are symmetric, then so is A+B. 30. ProveTheorem.5 for A an arbitrary2 3matrix and B any 3 matrix. 3. An n n matrix is upper triangular if a ij = 0 when i > j, that is, all the terms below the main diagonal are zero. Show that the sum and product of two upper triangular matrices of the same size are also upper triangular. 32. A matrix is said to be lower triangular if its transpose is upper triangular. Show that a matrix is lower triangular if and only if a ij = 0 when i < j. Show that the sum and product of two lower triangular matrices are also lower triangular. 33. Show that AA T and A T A are symmetric for any matrix A. 34. Prove Theorem Let 2 A = B = ] C = Show that AB = AC. Thus, it is possible for AB = AC with A different from the zero matrix and B not equal to C.

21 .3. ELEMENTARY ROW OPERATIONS 2.3 Elementary Row Operations In this section we discuss in more detail the basic ideas involved in Gaussian elimination, and how one systematically reduces a matrix to the form (.6). The operations used in Gaussian elimination are called the elementary row operations. There are three of them which we list below. Definition.8. The three elementary row operations are:. Interchange any two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of one row to another. The following example illustrates these three operations. Example. Let A = a. Interchange rows and 2 of A: R R b. Multiply row 3 of A by 8 : 8 R c. Add four times the second row of A to the first row. 4R 2 +R Definition.9. Let A and B be two m n matrices. We say that B is row equivalent to A if there is a sequence of elementary row operations that transforms A into B.

22 22 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS Example 2a. Let A = 2 4. Show that I 3 2 = R R R +R R R 2 +R is row equivalent to A. 0 Thus, the sequence of row operations R R 2, 2R + R 2, 2 R 2, and ( 3R 2 +R ) performed in the given order transforms A into I 2. Example 2b. Show that the matrix A above is row equivalent to I 2. Solution. We know there is a sequence of elementary row operations that transforms A into I 2, and we expect that the row operations that undo the effect of these operations should then transform I 2 into A. The operations we used above and their inverses are R R 2 R R 2 2R +R 2 2R +R 2 2 R 2 2R 2 3R 2 +R 3R 2 +R Performing the row operations in the second column in reverse order upon I 2 we have 0 3 3R 2 +R R R +R R R 2 = A The previous example is a particular case of a general fact. Lemma.. Given any sequence of elementary row operations that transforms A into B, there is another sequence that transforms B into A. This is the same as saying that if B is row equivalent to A, then A is row equivalent to B.

23 .3. ELEMENTARY ROW OPERATIONS 23 Proof. We first show that each elementary row operation has a corresponding elementary row operation that cancels the effect of the first operation. Thus, let A be an arbitrary m n matrix. Suppose R j R k A B That is, rows j and k of A are interchanged, and the resulting matrix is B. Clearly if we interchange the same two rows, then we transform B back into A. We also have: If cr j : A B, then (/c)r j : B A If cr j +R k : A B, then cr j +R k : B A Now suppose B is row equivalent to A. Then there is a sequence of elementary row operations E,...,E m which when performed in the given order, that is, E m followed by E m,..., followed by E, transforms A into B. Let F j denote the inverse of E j. Thus, if E is 2R 3, then F is 2 R 3. Then F m,...,f when performed on B, in this order, will transform B into A; cf. Example 2. The following theorem lists some properties of row equivalence: Theorem.6. Let A,B, and C be m n matrices, Then: a. A is row equivalent to itself. b. If A is row equivalent to B, then B is row equivalent to A. c. If A is row equivalent to B and B is row equivalent to C, then A is row equivalent to C. Proof. Partaisobvious,andbisLemma.. Toseethatcistrue, lete,...,e p denote a sequence of elementary row operations that transform A into B. Let F,...,F q denote a sequence of elementary row operations that transform B intoc. Clearlythe followingsequenceofrowoperations, F,...,F q, E,...,E p, transform A into C. Properties a, b, and c are called reflexivity, symmetry, and transitivity, respectively. A more explicit description of what we meant by (.6) is: Definition.0. The matrix A is said to be in row echelon form if it satisfies the following conditions:. All rows consisting entirely of zeros must be at the bottom of the matrix; i.e., no row of zeros may precede any row that has a nonzero entry. 2. The first nonzero entry of any row must be the number. 3. All entries directly below the first nonzero entry of any row must be zero. 4. The number of the column containing the first nonzero entry of a row must increase as we move down the rows of the matrix.

24 24 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS Example 3. 0 a. The matrix is not in row echelon form since the first nonzero entry in row2is not a. If row2is divided by 3, we get the rowequivalent matrix which is in row echelon form b. is not in row echelon form but the matrix which comes from the preceding matrix after a row interchange is in row echelon form. c. 0 0 is not in row echelon form, since there is a nonzero entry in the 3,2 position, which is directly below the first nonzero entry of row 2. d is not in row echelon form, since condition 4 of Definition is not satisfied by this matrix. The first nonzero terms do not move to the right as we go down the rows. Definition.. A matrix is said to be in reduced row echelon form if it is in row echelon form and also satisfies: 5. All entries above the first nonzero entry of any row are zero. Example 4. 0 a. is in row echelon form, but not in reduced row echelon form. [ 0 0 ] 0 0 is in reduced row echelon form. 0 0 b is in neither row echelon form nor reduced row echelon form. 0 Condition 3 does not hold for this matrix. c is in reduced row echelon form

25 .3. ELEMENTARY ROW OPERATIONS 25 We next state two theorems but do not give their proofs. Theorem.7. Every matrix is row equivalent to a unique matrix in reduced row echelon form. Theorem.8. Two matrices are row equivalent if and only if they are row equivalent to the same matrix in reduced row echelon form. In general if we wish to determine whether or not two matrices are row equivalent, we use row operations to find the reduced row echelon form they are row equivalent to, and then compare these two forms. Example 5. Show that A = is row equivalent to the identity 0 0 matrix. To do this, one performs row operations on A until it is transformed into the 3 3 identity matrix R 3 +R R 3 +R R 2 followed by 2R 2 +R gives R Interchanging R and R 2, we have Example 6. Show that B = is not row equivalent to the matrix 0 0 A of Example 5. To do this, we show that B is row equivalent to a matrix that

26 26 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS cannot be row equivalent to I 3. We deduce from this, via Theorem.8, that A and B cannot be row equivalent R 3 +R R 2 +R R Since the last matrix is in reduced row echelon form, and I 3 is also in reduced row echelon form, Theorem.7 implies B cannot be row equivalent to I 3, and hence B cannot be row equivalent to A. For if B were row equivalent to A, then by Theorem.8 and Example 3, B would be row equivalent to I 3. Actually there was no need to perform any row operations on B at all, since it is clear from the structure of B that no sequence of row operations can put a in its second column. For each of the elementary row operations there is a corresponding elementary column operation. For the most part, however, when solving systems of linear equations, only the elementary row operations are used. The reader should also know that each of the elementary row (column) operations has an elementary row (column) matrix associated with it. For example, if we are dealing with 3 n matrices, then the elementary row matrix associated with a particular elementary row operation is that matrix which results when the given elementary row operation is performed upon I 3.. R R 2 : cr 2 : c cr 2 +R 3 : c These examples demonstrate the relationship between the elementary row operation and its associated matrix. If we are in the class of m n matrices (3 3 above), we would start out with I m ; i.e., the number of columns is irrelevant.

27 .3. ELEMENTARY ROW OPERATIONS 27 Example 7. Let A be any 3 3matrix. Let E be the elementary matrix associated with 2R +R 3. Show that EA equals the matrix obtained by transforming A with this row operation. Solution. The matrix E, obtained by transforming I 3 with 2R +R 3, equals Computing EA for an arbitrary 3 3 matrix, we have EA = a b c d e f = a b c d e f 2 0 g h i 2a+g 2b+h 2c+i and 2R +R 3 : a b c d e f a b c d e f = EA g h i 2a+g 2b+h 2c+i The equality demonstrated in this example is not a coincidence. It is true in general, and the reader is asked to show this in the problems at the end of this section. Example 8. Let A = Let E = Show that the first column of the matrix product EA equals [ ] T. Note that E is the matrix product 0 0 E = Thatis, E istheresultofapplyingthetwoelementaryrowoperations 2R +R 2 and 5 2 R +R 3 consecutively to I EA = = Thus, EA equals the matrix that results from applying the above two row operations to A. Problem Set.3. Which of the following matrices are in row echelon form?

28 28 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS a. b ,,, , , Which of the following matrices are in reduced row echelon form? a.,,, b , , c , Find the reduced row echelon form to which each of the following matrices is row equivalent. 6 2 a., b , c Find the reduced row echelon form to which each of the following matrices is row equivalent a. b Find the reduced row echelon form of each of the following matrices: 2 a b. [ ] 4 c d For each of the following elementary row operations, write the 3 3 elementary row matrix that corresponds to it. a. 2R 2 +R 3 b. R R 3 c. 3R 2

29 .3. ELEMENTARY ROW OPERATIONS For each of the following sequences of row operations find a 2 2 matrix E such that, if A is any 2 3 matrix, EA equals the matrix derived from A after the given sequence of row operations is performed on A. a. R R 2,2R 2 +R b. R +R 2,2R +R 2,3R 2 +R 0 8. Let E be the matrix. We know that if A is any 2 2 matrix 0 then EA is the matrix we get after interchanging the first two rows of A. Describe EA in terms of A Let E =. Let A be any 2 2 matrix. What column operations 0 performed upon A will produce the product matrix AE? 0. If E is the m m elementary matrix that corresponds to multiplying row i by c, then EA is the matrix we get when row i of A is multiplied by c. Show this for m = 2 or 3. How is AE related to A?. If E is the m m elementary matrix that corresponds to interchanging row i with row j, then EA is the matrix we get when rows i and j of A are interchanged. Show this for m = 2 or 3. How is AE related to A? 2. An n n matrix D = [d ij ] is said to be a diagonal matrix if d ij = 0 whenever i j. That is, [ the only ] nonzero entries of D are on the main 0 diagonal. Let D equal. For each of the following matrices, A, 0 2 compute DA and AD. a b c Let D = d d 2 0, E = e e 2 0. Compute DE. 0 0 d e 3 4. Let A be an arbitrary3 3matrix, and let D be any 3 3diagonalmatrix. Describe the rows of DA and the columns of AD. 5. Let A = [a jk ] be any 3 3 matrix for which a is not zero. Let E be the 0 0 matrix a 2 /a 0. Describe the first column of EA. a 3 /a 0 6. How many different 2 2 matrices in reduced row echelon form are there? Find at least four.

30 30 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS a b a+c b+d 7. LetA =. SupposeE isa2 2matrixsuchthatEA =. c [ d ] c d Clearly E = is one such matrix. Are there any others? 0 8. For which 2 2 diagonal matrices D is there at least one nonzero matrix A such that DA is the 2 2 zero matrix? 0 9. Find all 2 2 matrices A such that AB is symmetric, where B = Matrices and Linear Equations In this section we show how to write a system of linear equations as a matrix equation, and then demonstrate that the elementary row operations do not change the solution set. Let s take a simple system of linear equations and rewrite it as an equation involving matrices. 2x +6x 2 = 5 (.0) Let A = 2 6 and let X = 3 AX = x +3x 2 = 4 [ x x 2 ]. Then we can write (.0) as [ 2 6 x 5 = 3 x 2 4] Now let s write the general linear system of m equations in n unknowns: (.) a x +a 2 x 2 + +a n x n = b a 2 x +a 22 x 2 + +a 2n x n = b 2... a m x +a m2 x 2 + +a mn x n = b m (.2) as a matrix equation. Let A = [a ij ],X = [x,...,x n ] T, and B = [b,...,b m ] T. The matrix A will be referred to as the coefficient matrix of (.2) and X as the unknown solution If some of the b j are nonzero, B is called the nonhomogeneous term. We see, keeping in mind the definitions of matrix multiplication and equality, that (.2) can be written as AX = B (.3) Another matrix associated with (.2) is a a 2... a n b A = a 2 a a 2n b 2... (.4) a m a m2... a mn b m

31 .4. MATRICES AND LINEAR EQUATIONS 3 It is called the augmented matrix of (.2). Notice that the augmented matrix A is just the coefficient matrix A with the right-hand side of (.2) added on as an (n+)st column. We want to show that if we have a system of equations and change it to a second system by performing some elementary row operations, we have neither lost nor gained any solutions. That is, the solution sets for the two systems are the same. Theorem.9. Let A and B be the augmented matrices of two systems of linear equations. If A and B are row equivalent, the two systems have the same solutions sets. Proof. SinceAandB arerowequivalentonlyifthereisasequenceofelementary row operations that transforms A into B, it suffices to prove this theorem in the case where only one of the three elementary row operations is used. Case. Suppose two rows of A, when interchanged, yield B. Clearly all that we ve done is to write the equations in a different order. Hence the solutions to both of these systems must coincide. Case 2. Some row of A when multiplied by a nonzero constant yields B. Here we ve multiplied one of the equations of the first system by a nonzero constant. This too leaves the solution set invariant. Case 3. Suppose cr j is added to R k ; that is, the kth row of B is c times the jth row of A added to the kth row of A. Suppose the numbers x,x 2,...,x n satisfy the first system. In particular this means a j x + +x jn x n = b j a k x + +a kn x n = b k Adding c times the first equation to the second equation gives us the following pair of equations: a j x + + a jn x n = b j (ca j +a k )x + +(ca jn +a kn )x n = cb j +b k But the coefficients of these two equations are the jth and kth rows of B, respectively. Since the remaining m 2 rows of A and B are equal, we see that these n numbers x i also satisfy the second system. Moreover since cr j +R k when performed upon B will yield A, a similar argument shows that every solution to the second system is also a solution to the first system Example. Solve the following system of equations by finding a matrix in row echelon form which is row equivalent to the augmented matrix. 2x x 2 + x 4 = x 2 +x 3 +x 4 = 0 x +x 2 +x 3 = 6

32 32 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS The augmented matrix is and it is row equivalent to the following matrices: R R R +R R 2 +R The above matrix is in row echelon form. The system of equations for which it is the augmented matrix has as solutions: x 3 = 3 4x 4 x 2 = 3+3x 4 x = 6+ x 4 (.5) If a numerical value for x 4 is specified, setting x,x 2, and x 3 equal to those numbers which satisfy (.5) gives us a solution to the original system of equations. Moreover, every solution may be generated in this manner. We obtain two explicit solutions by setting x 4 equal to 0 and then : (6,3, 3,0) (7,6, 7,) As usual let A = [a ij ] and X = [x,...,x n ] T be m n and n matrices, respectively. Let B = [b,...,b m ] T be an m matrix. Definition.2. A system of equations AX = B is said to be homogeneous if B = O m, i.e., b = 0, b 2 = 0,...,b m = 0. The zero solution to a homogeneous system is called the trivial solution. If B O m, the system is said to be nonhomogeneous. The following result, which is almost obvious, is extremely useful. Theorem.0. Let A and B be as above. If X p is any solution to AX = B, every other solution to AX = B can be written in the form X = H+X p, where H is a solution to the associated homogeneous equation AX = O m. Proof. Suppose AX = B. Set H = X X p. Clearly X = X p + (X X p ) = X p +H. Moreover AH = A(X X p ) = AX AX p = B B = O m.

33 .4. MATRICES AND LINEAR EQUATIONS 33 Note also that if AH = O m, then A(X p +H) = AX p +AH = B +O m = B. Example 2. Write the solutions to the system of equations in Example in the form H +X p. Solution. From (.5) one particular solution (obtained by setting x 4 equal to zero), is (6,3, 3,0). Thus, we have 6 X = 3 3 +x = X p +H 0 where H equals x 4 (,3, 4,) T. It is easy to show that, no matter what value x 4 is given, H is a solution to the homogeneous system. Later we will need to know that every homogeneous system of equations with more unknowns than equations always has a nontrivial solution, that is, a solution for which not all the x j s are zero. Before proving this, we look at a specific example. Example 3. Show that the following system has a nontrivial solution (two equations three unknowns). 3x +2x 2 = 0 2x + x 2 x 3 = 0 Solution. Writing the coefficient matrix, we have This matrix is row equivalent to This matrix implies that x = 2 7 x 3 and x 2 = 3 7 x 3 are the only solutions to the system. Clearly there are many nontrivial solutions. One is obtained by setting x 3 equal to, another by setting x 3 equal to 2, etc. Theorem.. Let A = [a ij ] be an m n matrix with m < n (more unknowns than equations). Then there is a nontrivial solution to the homogeneous equation AX = O m. Proof. We prove this by induction on the number of equations. Suppose first that m =, i.e., only one equation. a x +a 2 x 2 + +a n x n = 0 (.6)

34 34 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS Since we haveanequation, at least one ofthe coefficients a j is not zero. Suppose a 0. Set x 2 = and x 3 = = x n = 0. Setting x = a 2 /a gives us a nontrivial solution to (.6). Now suppose that the theorem is true for p equations, p < m. Using this we show that the theorem must be true for m equations. Let A be the matrix in reduced row echelon form that is row equivalent to A. To show that AX = O m has a nontrivial solution it suffices to show that A X = O m has a nontrivial solution. Case. Assume A has no row of zeros. Then there are exactly m columns with the number as the only nonzero entry. Let x i,x i2,...,x im, i < i 2 < < i m n be the unknowns associated with these columns. Let k be any integer, k n, not equal to any of the i j, j m. Set x k =. Set all the other variables, excepting the x ij, equal to 0. Solve the resulting system for the remaining m unknowns x ij. Since x k =, we have a nontrivial solution to our homogeneous system. Case 2. A has a row of zeros. This means that A X = O m is actually a system of p equations with p < m. By our induction hypothesis this system has a nontrivial solution. We conclude this section with two examples of how matrices may arise from different problems. Example 4. Suppose that a rabbit store starts out with 5 male rabbits and 4 female rabbits. Suppose also that every 6 months each female rabbit gives birth to 3 male rabbits and 3 female rabbits. How many male and female rabbits will there be (assuming no deaths) after 2 years? Solution. We need to express how many males and females we have during any 6-month time period. Let m k and f k represent the number of males and females, respectively, during the kth time period. Then the following equations describe how the rabbit population grows from one time period to the next: m k+ = m k +3f k f k+ = f k +3f k Thus mk+ 3 mk = f k f k 3 Let A be the matrix. We have 0 4 mk+ mk = A = A f k+ f 2 mk k f k = A k+ 5 4 = = A k+ m0 f 0

35 .4. MATRICES AND LINEAR EQUATIONS 35 For a proof of this by induction see Appendix A. Thus, after 2 years or four time periods, we have m4 5 = A f The reader is asked to complete this example by computing A 4. Example 5. Suppose we have some system that can be in any one of three different states, which are denoted by S,S 2, and S 3. The system is such that it can pass directly from some states to other states, as Figure.2 indicates. The diagram tells us that from S the system can either stay in S or enter S 2. From S 3 S 3 S S 2 Figure.2 it must enter S or S 2, and once the system is in S 2 it cannot leave. We catalog these data in a matrix T = [t jk ]. T is often referred to as a transition matrix. The t jk are defined by t jk =, if the system can pass directly from state j to k. t jk = 0, if the system cannot pass directly from state j to state k. Thus T equals Notice that the third column has all zeros; this is because the system can never enter the third state. We note too that each row must have at least one nonzero entry. The positive powers of the transition matrix T contain information about the system. For example T 2 = [t 2jk ] equals where t 2jk = t j t k +t j2 t 2k +t j3 t 3k Now t jp t pk = or 0, and it equals only if the system can pass from state S j to state S k by going through state S p first. Thus t 2jk counts the number of

36 36 CHAPTER. MATRICES AND SYSTEMS OF LINEAR EQUATIONS ways that the system can pass from S j to S k with one intermediate state. In particular t 22 = t t 2 +t 2 t 22 +t 3 t 2 = ()()+()()+(0)() = 2 Hence the system can pass from S to S 2 with one intermediate state (both S and S 2 are allowed to be used as intermediate states) in two different ways. A similar analysis shows that the j,k entry of the matrix T n counts the number of ways the system can pass from state S j to state S k with n intermediate states. We note that any of the n possible states can function as an intermediate state. For our particular T we have T n = n n 0 Thus, there must be three different ways in which the system can pass from state S to S 2 with two intermediate stops, etc. Problem Set.4. For each of the following systems of equations, write the corresponding coefficient matrix and augmented matrix, and then determine the solution sets. a. 0x x 2 + x 3 = 9 x +6x 2 = 4 x +5x 2 7x 3 = b. 3x 2 +7x 3 x = x 3 + x x 2 = 4 x x 2 x 3 = 5 c. x +x 2 = x 2 +x 3 = 2 x 3 +x 4 = 3 x 4 +x 5 = 4 x +x 5 = 5 2. Assume each of the following matrices is the coefficient matrix for a system of homogeneous equations. Write the system and find all solutions a. b. c

37 .4. MATRICES AND LINEAR EQUATIONS Assume each of the matrices in problem 2 is the augmented matrix of a system of equations. Write out the system and solve if possible. 4. Assume each of the following matrices is the augmented matrix of a system of linear equations. Write the system and then find all solutions a. b Assume each of the matrices in problem 4 is the coefficient matrix of a homogeneous system. What are the systems? What are the solution sets? 6. Suppose that you have money in two separate investments, I and I 2. Assume I returns 0 percent per year and I 2 5 percent per year. At the end of each year half of the money you made in I you reinvest in I, and the other half you invest in I 2, alwaysleaving the originalamount invested where it is. Suppose also that any monies made in I 2 are left there. If you invest $000 in I and $2000 in I 2, how much will be invested in each account during the second year? During the third year? 7. This is a continuation of problem 6. If a k and a 2k represent the amount of money invested in accounts I and I 2, respectively, during the kth year, find a formula that gives a (k+) and a 2(k+) in terms of a k and a 2k. See if you can describe a k and a 2k in terms of a and a 2, i.e., in terms of the original investment. a 0 8. Let B =. b [ ] a. Show that B 4 a = 4 0 a 3 +a 2 b+ab 2 +b 3 b 4. a n 0 b. Show that B n = n. a n k b k b n k=0 Notice that if a = b we have B n = a n 0 na n a n 9. The following diagrams describe the transition properties of three different systems. Determine the transition matrix for each system and then calculate the nth power of each matrix. 0. The following diagram describes the transition properties of a four-state system. a. Determine the transition matrix T for this system.