7.2 Matrix Algebra. DEFINITION Matrix. D 21 a 22 Á a 2n. EXAMPLE 1 Determining the Order of a Matrix d. (b) The matrix D T has order 4 * 2.

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1 530 CHAPTER 7 Systems and Matrices 7.2 Matrix Algebra What you ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications... and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices. Matrices A matrix is a rectangular array of numbers. Matrices provide an efficient way to solve systems of linear equations and to record data. The tables of data presented in this textbook are examples of matrices. DEFINITION Matrix Let m and n be positive integers. An m : n matrix (read m by n matrix ) is a rectangular array of m rows and n columns of real numbers. a a 2 Á a n a D 2 a 22 Á a 2n T o o o a m a m2 Á a mn We also use the shorthand notation 3a ij 4 for this matrix. Each element, or entry, a ij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a ij is in the ith row and jth column. In general, the order of an m : n matrix is m * n. If m = n, the matrix is a square matrix. Two matrices are equal matrices if they have the same order and their corresponding elements are equal. EXAMPLE Determining the Order of a Matrix (a) The matrix c -2 3 has order 2 * d (b) The matrix D T has order 4 * (c) The matrix C 4 5 6S has order 3 * 3 and is a square matrix Now try Exercise. Matrix Addition and Subtraction We add or subtract two matrices of the same order by adding or subtracting their corresponding entries. Matrices of different orders cannot be added or subtracted. Historical Note Methods used by the Chinese between 200 B.C.E. and 00 B.C.E. to solve problems involving several unknowns were similar to modern methods that use matrices. Matrices were formally developed in the 8th century by several mathematicians, including Leibniz, Cauchy, and Gauss. DEFINITION Matrix Addition and Matrix Subtraction Let A = 3a ij 4 and B = 3b ij 4 be matrices of order m * n.. The sum A B is the m * n matrix A + B = 3a ij + b ij The difference A B is the m * n matrix A - B = 3a ij - b ij 4.

2 SECTION 7.2 Matrix Algebra 53 EXAMPLE 2 Using Matrix Addition Matrix A gives the mean SAT verbal scores for the six New England states over the time period from 200 to (Source: The College Board, World Almanac and Book of Facts, 2005.) Matrix B gives the mean SAT mathematics scores for the same 4-year period. Express the mean combined scores for the New England states from 200 to 2004 as a single matrix. A = CT ME MA NH RI VT F V B = CT ME MA NH RI VT F V SOLUTION The combined scores can be obtained by adding the two matrices: Power of Matrix Algebra The result in Example 2 is fairly simple, but it is significant that we found (essentially) 24 pieces of information with a single mathematical operation. That is the power of matrix algebra. A + B = CT ME MA F V NH RI VT Now try Exercise. When we work with matrices, real numbers are scalars. The product of the real number k and the m * n matrix A = 3a ij 4 is the m * n matrix ka = 3ka ij 4. The matrix ka = 3ka ij 4 is a scalar multiple of A. EXAMPLE 3 Using Scalar Multiplication A consumer advocacy group has computed the mean retail prices for brand name products and generic products at three different stores in a major city. The prices are shown in the 3 * 2 matrix. Store A Store B Store C Brand Generic C S The city has a combined sales tax of 7.25%. Construct a matrix showing the comparative prices with sales tax included. SOLUTION Multiply the original matrix by the scalar.0725 to add the sales tax to every price. Brand Generic Store A * C S L Store B C S Store C Now try Exercise 3.

3 532 CHAPTER 7 Systems and Matrices Matrices inherit many properties possessed by the real numbers. Let A = 3a ij 4 be any m * n matrix. The m * n matrix O = 304 consisting entirely of zeros is the zero matrix because A + O = A. In other words, O is the additive identity for the set of all m * n matrices. The m * n matrix B = 3-a ij 4 consisting of the additive inverses of the entries of A is the additive inverse of A because A + B = O. We also write B = -A. Just as with real numbers, A - B = 3a ij - b ij 4 = 3a ij + -b ij 24 = 3a ij b ij 4 = A + -B2. Thus, subtracting B from A is the same as adding the additive inverse of B to A. EXPLORATION Computing with Matrices Let A = 3a ij 4 and B = 3b ij 4 be 2 * 2 matrices with a ij = 3i - j and b ij = i 2 + j 2-3 for i =, 2 and j =, 2.. Determine A and B. 2. Determine the additive inverse -A of A and verify that A + -A2 = 304. What is the order of 304? 3. Determine 3A - 2B. Matrix Multiplication To form the product AB of two matrices, the number of columns of the matrix A on the left must be equal to the number of rows of the matrix B on the right. In this case, any row of A has the same number of entries as any column of B. Each entry of the product is obtained by summing the products of the entries of a row of A by the corresponding entries of a column of B. DEFINITION Matrix Multiplication Let A = 3a ij 4 be an m * r matrix and B = 3b ij 4 an r * n matrix. The product AB = 3c ij 4 is the matrix where c ij = a i b j + a i2 b 2j + Á m * n + a ir b rj. The key to understanding how to form the product of any two matrices is to first consider the product of a * r matrix A = 3a j 4 with an r * matrix B = 3b j 4. According to the definition, AB = 3c is the matrix where c = a b + a 2 b 2 + Á 4 * + a r b r. For example, the product AB of the * 3 matrix A and the 3 * matrix B, where is A # B = # C 4 A = and B = C 5 S, S = 3 # # # 64 = Then, the ij-entry of the product AB of an m * r matrix with an r * n matrix is the product of the ith row of A, considered as a * r matrix, with the jth column of B, considered as an r * matrix, as illustrated in Example 4.

4 SECTION 7.2 Matrix Algebra 533 EXAMPLE 4 Finding the Product of Two Matrices Find the product AB if possible, where (a) (b) A = c d -3 A = c2 0 2 d and and -4 B = C 0 2S. 0 B = c d. [A] [B] [[ 6] [2 2 ]] FIGURE 7.7 The matrix product AB of Example 4. Notice that the grapher displays the rows of the product as * 2 matrices. SOLUTION (a) The number of columns of A is 3 and the number of rows of B is 3, so the product AB is defined. The product AB = 3c ij 4 is a 2 * 2 matrix where Thus, c = C 0-4 c 2 = C 2 0 c 2 = C 0-4 c 22 = C AB = c 2 2 d. S = 2 # + # # = -, S = 2 # # ) # 0 = -6, S = 0 # + # # = 2, S = 0 # # # 0 = 2. Figure 7.7 supports this computation. (b) The number of columns of A is 3 and the number of rows of B is 2, so the product AB is not defined. Now try Exercise 9. EXAMPLE 5 Using Matrix Multiplication A florist makes three different cut flower arrangements for Mother s Day (I, II, and III), each involving roses, carnations, and lilies. Matrix A shows the number of each type of flower used in each arrangement. A = I II III C 6 6 7S The florist can buy his flowers from two different wholesalers (W and W2), but wants to give all his business to one or the other. The cost of the three flower types from the two wholesalers is shown in matrix B. B = Roses Carnations Lilies Roses Carnations Lilies W W C S Construct a matrix showing the cost of making each of the three flower arrangements from flowers supplied by the two different wholesalers. (continued)

5 534 CHAPTER 7 Systems and Matrices [A] T [B] [[ ] [ ] [ ]] FIGURE 7.8 The product A T B for the matrices A and B of Example 5. SOLUTION We can use the labeling of the matrices to help us. We want the columns of A to match up with the rows of B (since that s how the matrix multiplication works). We therefore switch the rows and columns of A to get the flowers along the columns. (The new matrix is called the transpose of A, denoted by A T.) We then find the product A T B: Rose Carn Lily W W2 W W2 I Rose I II C 8 6 3S * Carn C S = II C S III Lilly III Figure 7.8 shows the product A T B and supports our computation. Now try Exercise 47. Identity and Inverse Matrices The n * n matrix I n with s on the main diagonal (upper left to lower right) and 0 s elsewhere is the identity matrix of order n : n For example, If A = 3a ij 4 is any n * n matrix, we can prove (see Exercise 56) that that is, is the multiplicative identity for the set of n * n matrices. I n 0 0 Á Á 0 I n = E 0 0 Á 0 U. o o o o Á I 2 = c d, I = C 0 0S, and I 4 = D T AI n = I n A = A, If a is a nonzero real number, then a - = /a is the multiplicative inverse of a, that is, aa - = a/a2 =. The definition of the multiplicative inverse of a square matrix is similar. DEFINITION Inverse of a Square Matrix Let A = 3a ij 4 be an n * n matrix. If there is a matrix B such that AB = BA = I n, then B is the inverse of A. We write B = A - (read A inverse ). We will see that not every square matrix (Example 7) has an inverse. If a square matrix A has an inverse, then A is nonsingular. If A has no inverse, then A is singular.

6 SECTION 7.2 Matrix Algebra 535 [A] [B] [B] [A] [[ 0 ] [ 0 ]] [[ 0 ] [ 0 ]] FIGURE 7.9 Showing A and B are inverse matrices. (Example 6) EXAMPLE 6 Verifying an Inverse Matrix Prove that are inverse matrices A = c d and B = c - 3 d SOLUTION Figure 7.9 shows that AB = BA = I Thus, B = A - and A = B Now try Exercise 33. EXAMPLE 7 Showing a Matrix Has No Inverse Prove that the matrix A = c 6 3 is singular, that is, A has no inverse. 2 d SOLUTION Suppose A has an inverse B = c x y Then, AB = I 2. z w d. Using equality of matrices we obtain: AB = c dcx y z w d = c 0 0 d 6x + 3z 6y + 3w = c 2x + z 2y + w d = c 0 0 d 6x + 3z = 6y + 3w = 0 2x + z = 0 2y + w = Multiplying both sides of the equation 2x + z = 0 by 3 yields 6x + 3z = 0. There are no values for x and z for which the value of 6x + 3z is both 0 and. Thus, A does not have an inverse. Now try Exercise 37. Determinant of a Square Matrix There is a simple test that determines if a 2 * 2 matrix has an inverse. Inverse of a 2 * 2 Matrix If ad - bc Z 0, then c a b - c d d = ad - bc c d -b -c a d. In Exercise 55 we ask you to prove the theorem above. The number ad - bc is the determinant of the 2 * 2 matrix A = c a b and is denoted c d d a b det A = ` = ad - bc. c d` To define the determinant of a higher-order square matrix, we need to introduce the minors and cofactors associated with the entries of a square matrix. Let A = 3a ij 4 be an n * matrix. The minor (short for minor determinant ) M ij corresponding to the element na ij is the determinant of the n - 2 * n - 2 matrix obtained by deleting the row and column containing a The cofactor corresponding to is A ij = -2 i+j ij. M ij. a ij

7 536 CHAPTER 7 Systems and Matrices Computing Determinants We expect you to compute the determinant of a 2 * 2 matrix mentally, e.g., Example 8a: det A2 = 3*2 - *4 = 6-4 = 2. Using a grapher for higher-dimension matrices is appropriate, e.g., Example 8b: DEFINITION Determinant of a Square Matrix Let A = 3a ij 4 be a matrix of order n * n n The determinant of A, denoted by det A or ƒaƒ, is the sum of the entries in any row or any column multiplied by their respective cofactors. For example, expanding by the ith row gives det A = ƒaƒ = a i A i + a i2 A i2 + Á + a in A in. If A = 3a ij 4 is a 3 * 3 matrix, then, using the definition of determinant applied to the second row, we obtain a a 2 a 3 3 a 2 a 22 a 23 3 = a 2 A 2 + a 22 A 22 + a 23 A 23 a 3 a 32 a 33 = a a ` 2 a 3 ` + a a a ` a 3 ` 33 a 33 a 32 + a a ` a 2 ` a 32 = -a 2 a 2 a 33 - a 3 a a 22 a a 33 - a 3 a a 23 a a 32 - a 2 a 3 2 The determinant of a 3 * 3 matrix involves three determinants of 2 * 2 matrices, the determinant of a 4 * 4 matrix involves four determinants of 3 * 3 matrices, and so forth. This is a tedious definition to apply. Most of the time we use a grapher to evaluate determinants in this textbook, as shown in the margin note. a 3 a 3 EXPLORATION 2 Investigating the Definition of Determinant. Complete the expansion of the determinant of the 3 * 3 matrix A = 3a ij 4 started above. Explain why each term in the expansion contains an element from each row and each column. 2. Use the first row of the 3 * 3 matrix to expand the determinant and compare to the expression in. 3. Prove that the determinant of a square matrix with a zero row or a zero column is zero. We can now state the condition under which square matrices have inverses. THEOREM Inverses of n : n Matrices An n * n matrix A has an inverse if and only if det A Z 0. There are complicated formulas for finding the inverses of nonsingular matrices of order 3 * 3 or higher. We will use a grapher instead of these formulas to find inverses of square matrices.

8 SECTION 7.2 Matrix Algebra 537 EXAMPLE 8 Finding Inverse Matrices Determine whether the matrix has an inverse. If so, find its inverse matrix. (a) A = c d (b) 2 - B = C 2-3S - 0 det([b]) 0 [B] [[..2.5] [.5 0.5] [..2.5]] FIGURE 7.0 The matrix B is nonsingular and so has an inverse. (Example 8b) SOLUTION A = ad - bc = 3 # 2 - # 4 = 2 Z 0 (a) Since det, we conclude that A has an inverse. Using the formula for the inverse of a 2 * 2 matrix, we obtain A - = ad - bc c d -b -c a d = 2 c d -0.5 = c -2.5 d You can check that A - A = A - A = I 2. (b) Figure 7.0 shows that det B = -0 Z 0 and B - = C S You can use your grapher to check that B - B = BB - = I 3. Now try Exercise 4. We list five of the important properties of matrices, some of which you will be asked to prove in the exercises. Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.. Commutative property 2. Associative property Addition: Addition: A + B = B + A A + B2 + C = A + B + C2 Multiplication: Multiplication: (Does not hold in general) AB2C = ABC2 3. Identity property 4. Inverse property Addition: A + O = A Multiplication: order of A = n * n A # In = I n # A = A 5. Distributive property Multiplication over addition AB + C2 = AB + AC A + B2C = AC + BC Addition: A + -A2 = O Multiplication: order of A = n * n AA - = A - A = I n, ƒaƒ Z 0 Multiplication over subtraction AB - C2 = AB - AC A - B2C = AC - BC Applications Points in the Cartesian coordinate plane can be represented by * 2 matrices. For example, the point 2, -32 can be represented by the * 2 matrix We can calculate the images of points acted upon by some of the transformations studied in Section.5, using matrix multiplication as illustrated in Example 9.

9 538 CHAPTER 7 Systems and Matrices EXAMPLE 9 Reflecting with Respect to the x-axis as Matrix Multiplication Prove that the image of a point under a reflection across the x-axis can be obtained by multiplying by c d y y r α θ P x x SOLUTION The image of the point x, y2 under a reflection across the x-axis is x, -y2. The product 3x y4c 0 d = 3x -y4 0 - shows that the point x, y2 (in matrix form 3x y4) is moved to the point x, -y2 (in matrix form 3x -y4). Now try Exercise 57. FIGURE 7. Rotating the xy-coordinate system through the angle a to obtain the x y -coordinate system. (Example 0) Figure 7. shows the xy-coordinate system rotated through the angle a to obtain the x y -coordinate system. In Example 0, we see that the coordinates of a point in the x y -coordinate system can be obtained by multiplying the coordinates of the point in the xy-coordinate system by an appropriate 2 * 2 matrix. In Exercise 7, you will see that the reverse is also true. EXAMPLE 0 Rotating a Coordinate System Prove that the x, y 2 coordinates of P in Figure 7. are related to the x, y2 coordinates of P by the equations x =x cos a + y sin a y =-x sin a + y cos a. Then, prove that the coordinates x, y 2 can be obtained from the x, y2 coordinates by matrix multiplication. We use this result in Section 8.4 when we study conic sections. SOLUTION Using the right triangle formed by P and the x y -coordinate system, we obtain x =r cos u - a2 and y =r sin u - a2. Expanding the above expressions for x and y, using trigonometric identities for cos u - a2 and sin u - a2, yields x =r cos u cos a + r sin u sin a, and y =r sin u cos a - r cos u sin a. It follows from the right triangle formed by P and the xy-coordinate system that x = r cos u and y = r sin u. Substituting these values for x and y into the above pair of equations yields x =x cos a + y sin a and y =y cos a - x sin a = -x sin a + y cos a, which is what we were asked to prove. Finally, matrix multiplication shows that cos a -sin a 3x y 4 = 3x y4c sin a cos a d. Now try Exercise 7.

10 SECTION 7.2 Matrix Algebra 539 Chapter Opener Problem (from page 59) Problem: If we have a triangle with vertices at 0, 02,, 2, and 2, 02, and we want to double the lengths of the sides of the triangle, where would the vertices of the enlarged triangle be? Solution: Given a triangle with vertices at 0, 02,, 2, and 2, 02, as in Figure 7.2, we can find the vertices of a new triangle whose sides are twice as long by multiplying by the scale matrix. c d For the point 0, 02, we have 3x y 4 = 30 04c 2 0 d = For the point, 2, we have 3x y 4 = 3 4c 2 0 d = And for the point 2, 02, we have 3x y 4 = 32 04c 2 0 d = So the new triangle has vertices 0, 02, 2, 22, and 4, 02, as Figure 7.3 shows. y y (0, 0) (, ) (2, 0) x (0, 0) (2, 2) (4, 0) x FIGURE 7.2 FIGURE 7.3 QUICK REVIEW 7.2 (For help, go to Sections.5, 5.3, and 6.4.) In Exercises 4, the points (a) 3, -22 and (b) x, y2 are reflected across the given line. Find the coordinates of the reflected points.. The x-axis 2. The y-axis 3. The line y = x 4. The line y = -x 6. y P(x, y) r In Exercises 5 and 6, express the coordinates of P in terms of u. 5. y θ x 3 θ P(x, y) x In Exercises 7 0, expand the expression. 7. sin a + b2 8. sin a - b2 9. cos a + b2 0. cos a - b2

11 540 CHAPTER 7 Systems and Matrices SECTION 7.2 EXERCISES Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 6, determine the order of the matrix. Indicate whether the matrix is square c2 2. c 3. C - 2S 0 5 d - 2 d C - S In Exercises 7 0, identify the element specified for the following matrix C S a 3 8. a a a 33 In Exercises 6, find (a) A + B, (b) A - B, (c) 3A, and (d) 2A - 3B A = c - 5 d, B = c d A = C 4 - S, B = C - 0 2S A = C 0 - S, B = C -2 S A = c d, B = c d A = C S, B = C 0 S A = , and In Exercises 7 22, use the definition of matrix multiplication to find (a) AB and (b) BA. Support your answer with the matrix feature of your grapher A = c - 5 d, B = c d 8. A = c d, B = c d 9. A = c d, B = C B = S A = c d, B = D 0 2 T A = C 4 - S, B = C - 0 2S In Exercises 23 28, find (a) AB and (b) BA, or state that the product is not defined. Support your answer using the matrix feature of your grapher A = 32-34, B = C 4 S A = C 3 S, B = A = c - 2 d, B = A = D T, B = c d A = C 0 0S, B = C 2 0 S A = D T, B = D T In Exercises 29 32, solve for a and b. 29. c a -3-3 d = c b d 30. c - 0 a -2 d = c b d A = C -2 4S, B = C a C 2 3 S = C b + 2 3S S c a d = c4 2 0 b - d In Exercises 33 and 34, verify that the matrices are inverses of each other. 33. A = c d, B = c d A = C 2-2 S, B = C S

12 SECTION 7.2 Matrix Algebra 54 In Exercises 35 40, find the inverse of the matrix if it has one, or state that the inverse does not exist. 35. c c 2 2 d 0 5 d C 2-3S 38. C - 0 4S A = 3a ij 4, a ij = -2 i+j, i 4, j B = 3b ij 4, b ij = ƒi - jƒ, i 3, j 3 In Exercises 4 and 42, use the definition to evaluate the determinant of the matrix C - 0 2S 42. D T In Exercises 43 and 44, solve for X X + A = B, where A = c 3 d and B = c4 2 d X + A = B, where A = c d and B = c d. 45. Symmetric Matrix The matrix below gives the road mileage between Atlanta (A), Baltimore (B), Cleveland (C), and Denver (D). (Source: AAA Road Atlas) (a) Writing to Learn Explain why the entry in the ith row and jth column is the same as the entry in the jth row and ith column. A matrix with this property is symmetric. (b) Writing to Learn Why are the entries along the diagonal all 0 s? A B C D D T Production Jordan Manufacturing has two factories, each of which manufactures three products. The number of units of product i produced at factory j in one week is represented by in the matrix If production levels are increased by 0%, write the new production levels as a matrix B. How is B related to A? 47. Egg Production Happy Valley Farms produces three types of eggs: (large), 2 (X-large), 3 (jumbo). The number of dozens of type i eggs sold to grocery store j is represented by in the matrix. a ij a ij A B C D A = C 50 0 S A = C S The per dozen price Happy Valley Farms charges for egg type i is represented by b i in the matrix $0.80 B = C $0.85 S. $.00 (a) Find the product B T A. (b) Writing to Learn represent? What does the matrix B T A 48. Inventory A company sells four models of one name brand all-in-one fax, printer, copier, and scanner machine at three retail stores. The inventory at store i of model j is represented by s ij in the matrix S = C S The wholesale and retail prices of model i are represented by p i and p i2, respectively, in the matrix $80 $ $275 $ P = D T. $355 $ $590 $ (a) Determine the product SP. (b) Writing to Learn represent? What does the matrix SP 49. Profit A discount furniture store sells four types of 5-piece bedroom sets. The price charged for a bedroom set of type j is represented by a j in the matrix A = 3$398 $598 $798 $9984. The number of sets of type j sold in one period is represented by b j in the matrix B = The cost to the furniture store for a bedroom set of type j is given by c j in the matrix C = 3$99 $268 $500 $6704. (a) Write a matrix product that gives the total revenue made from the sale of the bedroom sets in the one period. (b) Write an expression using matrices that gives the profit produced by the sale of the bedroom sets in the one period. 50. Construction A building contractor has agreed to build six ranch-style houses, seven Cape Cod-style houses, and 4 colonial-style houses. The number of units of raw materials that go into each type of house are shown in the matrix Steel Wood Glass Paint Labor Ranch R = Cape Cod C S. Colonial Assume that steel costs $600 a unit, wood $900 a unit, glass $500 a unit, paint $00 a unit, and labor $000 a unit. (a) Write a * 3 matrix B that represents the number of each type of house to be built.

13 542 CHAPTER 7 Systems and Matrices (b) Write a matrix product that gives the number of units of each raw material needed to build the houses. (c) Write a 5 * matrix C that represents the per unit cost of each type of raw material. (d) Write a matrix product that gives the cost of each house. (e) Writing to Learn Compute the product BRC. What does this matrix represent? 5. Rotating Coordinate Systems The xy-coordinate system is rotated through the angle 30 to obtain the x y - coordinate system. (a) If the coordinates of a point in the xy-coordinate system are, 2, what are the coordinates of the rotated point in the xy-coordinate system? (b) If the coordinates of a point in the x y -coordinate system are, 2, what are the coordinates of the point in the xy-coordinate system that was rotated to it? 52. Group Activity Let A, B, and C be matrices whose orders are such that the following expressions are defined. Prove that the following properties are true. (a) A + B = B + A (b) A + B2 + C = A + B + C2 (c) AB + C2 = AB + AC (d) A - B2C = AC - BC 53. Group Activity Let A and B be m * n matrices and c and d scalars. Prove that the following properties are true. (a) ca + B2 = ca + cb (b) c + d2a = ca + da (c) cda2 = cd2a (d) # A = A 54. Writing to Learn Explain why the definition given for the determinant of a square matrix agrees with the definition given for the determinant of a 2 * 2 matrix. (Assume that the determinant of a * matrix is the entry.) 55. Inverse of a 2 : 2 Matrix Prove that the inverse of the matrix A = c a b c d d is A- = ad - bc c d -b -c a d provided ad - bc Z Identity Matrix Let A = 3a ij 4 be an n * n matrix. Prove that AI n = I n A = A. In Exercises 57 6, prove that the image of a point under the given transformation of the plane can be obtained by matrix multiplication. 57. A reflection across the y-axis 58. A reflection across the line y = x 59. A reflection across the line y = -x 60. A vertical stretch or shrink by a factor of a 6. A horizontal stretch or shrink by a factor of c Standardized Test Questions 62. True or False Every square matrix has an inverse. Justify your answer. 63. True or False The determinant ƒaƒ of the square matrix A is greater than or equal to 0. Justify your answer. In Exercises 64 67, solve the problem without using a calculator. 64. Multiple Choice Which of the following is equal to the 2 4 determinant of A = c -3 - d? (A) 4 (B) -4 (C) 0 (D) -0 (E) Multiple Choice Let A be a matrix of order 3 * 2 and B a matrix of order 2 * 4. Which of the following gives the order of the product AB? (A) 2 * 2 (B) 3 * 4 (C) 4 * 3 (D) 6 * 8 (E) The product is not defined. 66. Multiple Choice Which of the following is the inverse of the matrix c d? 2 - (A) c -4 7 (B) c 2-7 (C) c -2 d - 4 d -7 4 d (D) c d (E) 67. Multiple Choice Which of the following is the value 2 3 of a 3 in the matrix 3a ij 4 = C 4 5 6S? (A) -7 (B) 7 (C) -3 (D) 3 (E) 0 Explorations 4-7 c - 2 d 68. Continuation of Exploration 2 Let A = 3a ij 4 be an n * n matrix. (a) Prove that the determinant of A changes sign if two rows or two columns are interchanged. Start with a 3 * 3 matrix and compare the expansion by expanding by the same row (or column) before and after the interchange. [Hint: Compare without expanding the minors.] How can you generalize from the 3 * 3 case? (b) Prove that the determinant of a square matrix with two identical rows or two identical columns is zero. (c) Prove that if a scalar multiple of a row (or column) is added to another row (or column) the value of the determinant of a square matrix is unchanged. [Hint: Expand by the row (or column) being added to.] 69. Continuation of Exercise 68 Let A = 3a ij 4 be an n * n matrix. (a) Prove that if every element of a row or column of a matrix is multiplied by the real number c, then the determinant of the matrix is multiplied by c. (b) Prove that if all the entries above the main diagonal (or all below it) of a matrix are zero, the determinant is the product of the elements on the main diagonal. 70. Writing Equations for Lines Using Determinants Consider the equation x y 3 x y 3 = 0. x 2 y 2

14 SECTION 7.2 Matrix Algebra 543 (a) Verify that the equation is linear in x and y. (b) Verify that the two points x, y 2 and x 2, y 2 2 lie on the line in part (a). (c) Use a determinant to state that the point x 3, y 3 2 lies on the line in part (a). (d) Use a determinant to state that the point x 3, y 3 2 does not lie on the line in part (a). 7. Continuation of Example 0 The xy-coordinate system is rotated through the angle a to obtain the x y -coordinate system (see Figure 7.). (a) Show that the inverse of the matrix A = c cos a -sin a sin a cos a d of Example 0 is A - cos a sin a = c -sin a cos a d. (b) Prove that the x, y2 coordinates of P in Figure 7. are related to the x, y 2 coordinates of P by the equations x = x cos a - y sin a y = x sin a + y cos a. (c) Prove that the coordinates x, y2 can be obtained from the x, y 2 coordinates by matrix multiplication. How is this matrix related to A? Extending the Ideas 72. Characteristic Polynomial Let A = 3a ij 4 be a 2 * 2 matrix and define ƒx2 = detxi 2 - A2. (a) Expand the determinant to show that ƒx2 is a polynomial of degree 2. (The characteristic polynomial of A) (b) How is the constant term of ƒx2 related to det A? (c) How is the coefficient of x related to A? (d) Prove that ƒa2 = Characteristic Polynomial Let A = 3a ij 4 be a 3 * 3 matrix and define ƒx2 = det xi 3 - A2. (a) Expand the determinant to show that ƒx2 is a polynomial of degree 3. (The characteristic polynomial of A) (b) How is the constant term of ƒx2 related to det A? (c) How is the coefficient of x 2 related to A? (d) Prove that ƒa2 = 0.