Algebra II Notes Matrices and Determinants Unit 04

Matrix Addition Big Idea: Matrices are used to organize, display, interpret, and analyze data. Algebraic operations are performed with matrices to solve real world problems. Matrices can be added and subtracted if and only if they have the same dimensions. Any matrix can be multiplied by a scalar reducing or enlarging the matrix. Objectives: N.VM.C.6 Perform operations on matrices and use matrices in applications. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. N.VM.C.7 Perform operations on matrices and use matrices in applications. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. N.VM.C.8 Perform operations on matrices and use matrices in applications. Add, subtract, and multiply matrices of appropriate dimensions. Skills: The student will organize data using matrices. The student will simplify matrix expressions using the properties of matrices. Vocabulary: Matrix: a rectangular arrangement of numbers (called entries) in rows (horizontal) and columns (vertical); plural of matrix: matrices Dimensions (Order) of a Matrix: Number of Rows (by) Number of Columns Square Matrix: a matrix with the same number of rows and columns Organizing Data in a Matrix Ex 1: Store A sells 550 DVDs, 420 video games, and 910 CDs on average every week. Store B sells 405 DVDs, 300 video games, and 1100 CDs on average every week. Use a matrix to organize this information. State the dimensions of this matrix. Solution: DVDs VGs CDs Store A 550 420 910 Store B 405 300 1100 The dimensions (order) of the matrix: 2 3 Adding & Subtracting Matrices: In order to add or subtract matrices, the dimensions must be the same. Ex 2: Add the matrices: 4 3 1 8 1 5. Step One: Determine if the matrices can be added by checking the dimensions. The matrices both have dimensions 1 3, so they can be added. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 1 of 49 4/3/2014

Step Two: Add the corresponding entries. The sum of the matrices will have the same dimensions as the original matrices. 4 3 1 8 1 5 4 8 3 1 1 5 12 4 4 Ex 3: Subtract the matrices: 3 0 12 5 9 1 5 8 6 0 13 7 Step One: Determine if the matrices can be added by checking the dimensions. The matrices both have dimensions 2 3, so they can be subtracted. 3 0 12 5 9 1 Step Two: Change to an addition problem (add the opposite). 5 8 6 0 13 7 Step Three: Add the matrices. 3 0 12 5 9 1 2 9 11 5 8 6 0 13 7 5 21 1 Scalar: another name for a real number (not a matrix) Scalar Multiplication Ex 4: Find the product. 6 1 0 2 3 3 4 2 5 Multiply every entry in the matrix by the scalar. 6 1 18 3 0 2 0 6 3 3 4 9 12 2 5 6 15 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 2 of 49 4/3/2014

Equal Matrices: two matrices are equal if their dimensions are the same, and all of the corresponding entries are equal. Ex 5: Solve the equation for x and y. 4 2x 5 1 9 13 8 1 5y 2 12 3 Step One: Add the matrices on the left side of the equation. 4 5 2x1 9 13 9 2x1 9 13 8 5y 1 2 12 3 8 5y 3 12 3 Step Two: Set the corresponding entries equal to each other and solve for the variables. 2x 1 13 2x 12 x 6 8 5y 12 5y 20 y 4 You Try: Perform the indicated operations. the order of operations! 7 0 2 1 0 8 6 3 2 4 6 3 5 Reminder: Be sure to use You Try: Library A has 10,000 novels, 5000 biographies, and 5000 children s books. Library B has 15,000 novels, 10,000 Biographies, and 2500 children s books. Library C has 4000 novels, 700 biographies, and 800 children s books. a. Express each library s number of books as a matrix. Label the matrices A, B, and C. b. Find the total number of each type of book in all 3 libraries. Express as a matrix. c. How many more books of each type does Library A have then Library C? d. Find A+B. Does the matrix have meaning in this situation? Explain. QOD: Is matrix addition commutative? Explain your answer. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 3 of 49 4/3/2014

Sample CCSD Common Exam Practice Question(s): 1. The tables show the number of students in band and choir by grade and gender. Students in Band Students in Choir Grade Girls Boys Grade Girls Boys 10 23 27 10 26 20 11 18 20 11 20 16 12 13 17 12 12 15 Which could be used to find the total number of girls and boys in the band and choir by grade level? A. B. C. D. 23 27 26 20 18 20 20 16 13 17 12 15 23 27 26 20 12 18 20 20 16 15 13 17 23 26 20 27 18 20 16 20 13 12 15 17 23 26 27 20 17 18 20 20 16 15 13 12 2. Find the values of x and y that make the equation true. 2x 2 4 8 16 6 3 6 10 7 y 1 A. x = 8, y = 3 B. x = 7, y = 3 C. x = 6, y = 13 D. x = 1, y = 1 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 4 of 49 4/3/2014

Sample Questions: 1. Write and label a 2 x 3 matrix to organize the following data about the t-shirt inventory at two stores. Stores A has 6 small, 19 medium, and 13 large t-shirts. Store B has 17 small, 34 medium, and 28 large t- shirts. b. d. ANS: D DOK 1 2. Two stores carry small, medium, and large sweatshirts. The table shows the inventory at the stores. Arrange the data in a matrix. Give the dimensions of the matrix. Sweatshirt Inventory Small Medium Large Store A 6 21 13 Store B 16 32 28 a. c. b. The dimensions are. d. The dimensions are. The dimensions are. ANS: D DOK 2 The dimensions are. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 5 of 49 4/3/2014

3. Add. + b. d. ANS: B DOK 1 4. Perform the indicated operation, if possible. If it is not possible, explain why not. b. d. Because the values of corresponding entries are different, the operation is not possible. ANS: B DOK 1 5. Perform the indicated operations, if possible. If they are not possible, explain. b. d. Because there is a zero in the second matrix, the operations are not possible. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 6 of 49 4/3/2014

ANS: A DOK 2 6. Evaluate B + C, if possible. Not possible b. d. ANS: A DOK 2 7. Katie asked the players on two ski teams what new color each team uniform should be: red, blue, or green. She recorded the results in two matrices. Find the total for the two teams. Males Females Males Females R B G R B G b. d. ANS: B DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 7 of 49 4/3/2014

8. The matrices show how many juniors and seniors at Douglass High School participated in three sports, football (F), soccer (S), and volleyball (V), over a three-year period. How many more juniors than seniors participated in soccer in the year 2003? Juniors Seniors a. 5 c. 105 b. 14 d. 7 ANS: A DOK 2 9. The matrix shows the number of runners from Central High (C) and West High (W) in four cross country races in 2003. In 2004 the number of runners in each race was four times as many as in 2003. Multiply the matrix by 4 to show the number of runners in 2004. Race b. d. ANS: C DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 8 of 49 4/3/2014

10. A movie theater marks up tickets by 120%. Use a scalar product to find the marked up prices. Movie Ticket Prices Time Child Adult Matinee $4 $6 Evening $6 $10 Late Night $5 $7 b. d. ANS: C DOK 2 11. Multiply the matrix of original prices by 1.20 and add to the matrix of original prices. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 9 of 49 4/3/2014

12. Represent the college tuition data shown in the table in a matrix. b. d. ANS: A DOK 1 13. At one supermarket, a bar of soap costs $1.60, a bottle of dishwashing liquid costs $4.00, and a bottle of furniture polish costs $3.00. These same items cost $2.40, $3.50, and $4.75, respectively, at another nearby market. Represent this information in the matrix. b. d. ANS: A DOK 1 14. Find the value of x. ANS: 4 DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 10 of 49 4/3/2014

17. Algebra II Notes Matrices and Determinants Unit 04 15. Solve for x. ANS: 2 DOK 2 16. Subtract: ANS: DOK 1 17. Perform the indicated matrix operation, if possible. ANS: DOK: 1 18. The Revenue and Expenses for two pet shops for a 2-month period are shown below. Write a matrix that shows the monthly profit for each pet shop. Which pet shop has the higher overall profit during the 2- month period? Revenue ($) Pets A Pets B Expenses ($) Pets A Pets B June July June July ANS: Profit ($) June Pets A Pets B July Pets A made more profit DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 11 of 49 4/3/2014

19. Use A( 4, 5), B(4, 5), and C( 6, 2). Write a matrix that represents the figure. ANS: DOK 1 20. A store manager orders orange and purple t-shirts. In April, he orders 7 small, 28 medium, 30 large, and 16 extra-large t-shirts in each color, as shown in the table. Small Medium Large Extra-Large Orange 7 28 30 16 Purple 7 28 30 16 Part A: Display this data in the form of a matrix A. Part B: The purple t-shirts sell out quickly. In May, the manager orders the same number of orange t- shirts in each size, but twice as many purple t-shirts. Display the data for May in a matrix M. ANS: DOK 2 Part A: Part B: 21. The table shows populations of an endangered species at three different locations in 2010. White River Laguna Grande Pleasant Valley 2010 1000 600 200 Part A: Display this data in the form of a matrix P. Part B: Assume that the population at White River increases 16% per year, the population at Laguna Grande decreases 20% per year, and the population at Pleasant Valley decreases 50% per year. Extend matrix P to show the populations for 2012 and 2013. If necessary, round populations to the nearest whole number. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 12 of 49 4/3/2014

Part C: Does the total population of the species increase, decrease, or stay the same from 2010 to 2012? Explain. ANS: DOK 3 Part A: Part B: Part C: The total population decreases slightly from 2010 to 2012. 2010 population: 1000 + 600 + 200 = 1800 2012 population: 1346 + 384 + 50 = 1780 22. Write a matrix which could represent the prices for small, medium, and large drinks at each of two different restaurants. ANS: DOK 2 Sample answer: Rest. Rest. 1 2 S M L 23. Write a matrix which could represent the prices for child and adult tickets at a movie theater that charges different prices for Saturday morning movies, for Saturday matinees, and for Saturday evening movies. ANS: DOK 2 Sample answer: Morn. Mat. Even. C A Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 13 of 49 4/3/2014

ESSAY 24. The payoff matrix shown reflects the values of 2 people in a situation where they either answer yes or no, and the respective values associated with those answers. They may change their answers as often as they wish. Person A is represented on the left of the matrix, and person B is represented in bold text along the top of the matrix, with values in bold text. Yes No Yes 1,1 0,3 No 3,2 4,1 Part A: Describe the tendency for each person to switch their answer from yes to no. Part B: A Nash equilibrium is a situation where neither person would benefit by changing answers. Does this situation model a Nash equilibrium? Explain why or why not. ANS: DOK 4 Part A: Analyze the situation when both people answer yes. The values for person A for an answer of yes are 1 (when person B answers yes ) and 0 (when person B answers no ), which change to 3 and 4 respectively if person A changes her answer. So person A should always change her answer. The values for person B for an answer of yes are 1 (when person A answers yes ) and 2 (when person A answers no ), which change to 3 and 1 respectively if person B changes her answer. So person B should change her answer only if person A answers yes. Part B: Yes. Person A will always answer no, because no always gives person A a greater payoff than yes. Person B will answer yes to get a value of 2 rather than 1. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 14 of 49 4/3/2014

Matrix Multiplication Big Idea: Before multiplying two matrices, it must be determined whether the matrix product is defined. Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. The associative and distributive properties apply to multiplication of square matrices, but the commutative property does not. Objectives: N.VM.C.8 Perform operations on matrices and use matrices in applications. (+) Add, subtract, and multiply matrices of appropriate dimensions. N.VM.C.9 Perform operations on matrices and use matrices in applications. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. Skill: The student will simplify matrix expressions using the properties of matrices. Matrix Multiplication: In order to multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The product matrix will have the same number of rows of the first matrix and the same number of columns of the second matrix. Let A be an mnmatrix, and B be an n p dimensions m p. matrix. The product AB exists, and will have the Ex 6: Find the product AB (if possible). 2 3 A 1 5 0 2 B 4 1 3 6 Step One: Determine if the product exists. If it does, find its dimensions. Matrix A is a 3 2 matrix. Matrix B is a 2 2 matrix. The number of columns in matrix A equals the number of rows in matrix B. Therefore, the product exists and will be a 3 2matrix. Step Two: Multiply each entry in the rows of matrix A to each entry in the columns of matrix B. Then find the sum of these products. 2 4 3 3 2 1 3 6 1 20 1 4 5 3 1 1 5 6 19 29 0 4 2 3 0 1 2 6 6 12 Ex 7: Find the product BA (if possible) of the matrices from the previous example. Step One: Determine if the product exists. If it does, find its dimensions. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 15 of 49 4/3/2014

Matrix B is a 2 2 matrix. Matrix A is a 3 2 matrix. The number of columns in Matrix B does not equal the number of rows in matrix A. Therefore, the product BA does not exist. Ex 8: Find the product AC if 1 2 7 1 3 3 A 4 5 2 0 8 4 and 6 C 8 2 (if possible). Matrix A is a 4 3 matrix. Matrix C is a 3 1 matrix. The number of columns in matrix A equals the number of rows in matrix C. Therefore, the product AC exists, and has the dimensions 4 1. 1 6 2 8 7 2 116 14 31 1 6 3 8 3 2 6 24 6 36 AC 4 6 5 8 2 2 24 40 4 12 0 6 8 8 4 2 0 64 8 72 Application Problem Multiplying Matrices Ex 9: A school is selling tickets to a school play. They sell tickets for $5 for balcony seating and $7 for floor seating. The school sells 60 balcony seats and 25 floor seats to parents and relatives. They also sell 40 balcony seats and 50 floor seats to students. Set up the school s profit as a product of two matrices. Seat Matrix Ticket Price Matrix = Profit Seat Matrix: Non-Student Tickets 60 25 Student Tickets 40 50 Ticket Price Matrix: Balcony Price5 Floor Price 7 Multiply the matrices: 60 25 5 60 5 25 7 Non-Student Profit 205 40 50 7 40 5 50 7 Student Profit 550 You Try: Find the product CA using the matrices 5 6 A 4 1 and 4 3 C 2 1. 0 8 QOD: Is matrix multiplication commutative? Explain why or why not. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 16 of 49 4/3/2014

Sample CCSD Common Exam Practice Question(s): If 6 2 A 1 4 0 5 and 4 2 5 B 4 6 1, which is the product AB? A. B. C. D. 26 25 18 37 22 9 30 21 16 24 28 20 22 20 20 30 5 32 0 32 12 26 1 20 30 5 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 17 of 49 4/3/2014

Sample Questions: 1. Tell whether the product of is defined. If so, give the dimensions of PQ. a. defined; c. defined; b. defined; d. undefined ANS: B DOK 2 2. Which statement is not true? a. The multiplication of three matrices is associative. b. The multiplication of three matrices is commutative. c. The addition of three matrices is associative. d. The addition of three matrices is commutative. ANS: B DOK 2 3. If P, Q, and R are matrices, which statement must be true? Why? a. P(QR) = P(RQ) because the commutative property holds for the multiplication of matrices. b. P(QR) = (PQ)R because the associative property holds for the multiplication of matrices. c. P(QR) = (QR)P because the commutative property holds for the multiplication of matrices. d. P(QR) = P(Q) + P(R) because the distributive property holds for the multiplication and addition of matrices. ANS: B DOK 3 4. If R, S, and T are matrices, which expression is equivalent to? Explain. a. RT + ST, because matrix multiplication is distributive. b. TR + TS, because matrix multiplication is distributive. c. Both RT + ST and TR + TS, because matrix multiplication is both commutative and distributive. d. Neither RT + ST nor TR + TS, because matrix multiplication is neither commutative nor distributive. ANS: B DOK 3 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 18 of 49 4/3/2014

5. Suppose J, K, and L are matrices. If, which of the following must be true? a. b. c. d. because matrix multiplication is associative. because matrix multiplication is associative. because matrix multiplication is commutative. because matrix multiplication is commutative. ANS: A DOK 3 6. Dana says Harry has multiplied in the wrong order and must redo the problem. Harry says the order he multiplies will not affect the product. Who is correct and why? a. Harry is correct because matrix multiplication is commutative. b. Dana is correct because matrix multiplication is not commutative. c. Harry is correct because matrix multiplication is associative. d. Dana is correct because matrix multiplication is not associative. ANS: B DOK 3 7. For the matrices, find the matrix products AB and BA and show that the products are not equal. b. d. ANS: D DOK 1 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 19 of 49 4/3/2014

8. Find the product AB, if possible. A = B = Not possible b. d. ANS: D DOK 2 9. Evaluate, if possible. a. Not possible c. b. d. ANS: D DOK 2 10. Given and, find AB. b. d. not possible ANS: A DOK: DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 20 of 49 4/3/2014

11. Given A = and B =, find AB. b. d. ANS: C DOK 2 12. Revenue and expenses for two bakeries for the first three days of last week are shown in the accompanying matrices. Write and label a matrix to show the daily profit for each bakery on those three days. Revenue Mon. Tues. Wed. Store 1 $64 $84 $91 Store 2 $97 $79 $69 Expenses Mon. Tues. Wed. Store 1 $62 $62 $51 Store 2 $72 $54 $76 b. d. The matrix operation cannot be performed. ANS: A DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 21 of 49 4/3/2014

13. Perform the indicated operations, if possible. If they are not possible, explain. b. d. Because there are values outside of each matrix, the operation is not possible. ANS: A DOK 2 14. July 2004 sales of car A and car B by two car dealers is shown in the first table. Use a product matrix to find the total profit from the cars for each dealer. July 2004 Car Sale Car A Car B Dealer 1 6 9 Dealer 2 12 10 Car Profits Revenue ($) Dealer Cost ($) Profit ($) Car A 8,700 2,900 5,800 Car B 12,800 3,200 9,600 a. The total profit from cars A and B for dealer 1 is $165,600 and for dealer 2 is $121,200. b. The total profit from cars A and B for dealer 1 is $121,200 and for dealer 2 is $165,600. c. The total profit from cars A and B for dealer 1 is $87,000 and for dealer 2 is $211,200. d. The total profit from cars A and B for dealer 1 is $104,400 and for dealer 2 is $148,200. ANS: B DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 22 of 49 4/3/2014

15. Student Government and the cheerleaders at a local school are ordering supplies. The supplies they need are listed below. If a bottle of paint costs $5, a roll of paper costs $12, and a roll of tape costs $2, which of the following shows the use of matrices to find the total cost of supplies for each group? a. b. c. d. ANS: C DOK 2 Short Answer: 16. A company stocks items A, B, and C at each of its two stores. Use matrix multiplication to determine the value of the inventory at each store. ANS: $198 at Store 1; $203 at Store 2 DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 23 of 49 4/3/2014

Determinants Big Idea: Determinants can be used to solve systems of equations. If the determinant is nonzero, then the system has a unique solution. If a determinant is 0, then the system either has no solution or infinite solutions. A method called Cramer s Rule uses the coefficient matrix. The coefficient matrix is a matrix that contains only the coefficients of the system. Cramer s Rule is used to solve a system of equations. Objectives: N.VM.C.9 Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. A.REI.C.8 Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable. A.REI.C.9 Solve systems of equations. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). Skills: The student will find the determinant of a matrix with and without technology. The student will solve systems of equations using matrices. Determinant of a Matrix Notation: Determinant of Matrix A = det A = A Evaluating the Determinant of a 22Matrix a b ad bc c d Ex 10: What is the determinant of the matrix A? 3 1 A 7 9 det A 39 17 27 7 20 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 24 of 49 4/3/2014

Evaluating the Determinant of a 3 3 Matrix a b c a b c a b det d e f d e f d e a e i b f g c d h b d i a f h c e g g h i g h i g h (Sum of the products of (Sum of the products of the diagonals left to right) the diagonals right to left) Ex 11: Evaluate the following: 5 1 2 0 3 4 1 4 2 5 1 2 5 1 0 3 4 0 3 532 1 4 1 2 0 4 1 02 54 4 2 31 1 4 2 1 4 =30 4 0 0 80 6 26 86 26 86 112 Application: Finding the Area of a Triangle Using Determinants The area of a triangle with vertices,,,, and, x y x y x y is 1 1 2 2 3 3 x1 y1 1 1 x2 y2 1. 2 x y 1 3 3 Note: The ± sign indicates that we need to choose which sign will make this a positive number. Ex 12: Find the area of a triangle whose vertices are the points 1,2, 3, 4, and 0,6. Area = 1 2 1 1 2 1 1 2 1 1 3 4 1 3 4 1 3 4 2 2 0 6 1 0 6 1 0 6 1 1 4 1 2 1 0 1 3 6 2 3 1 1 1 6 1 4 0 2 1 1 1 4 0 18 6 6 0 22 0 22 11 2 2 2 Solution: Because the determinant is positive, we will use the positive answer for the area of the triangle, 11 sq. units. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 25 of 49 4/3/2014

Solving a Linear System Using Cramer s Rule Linear System: ax by e cx dy f Coefficient Matrix: a c b d Constant Matrix: e f Cramer s Rule Solution of a Linear System: e b a e f d c f a b x and y, if 0 a b a b c d c d c d Note: Constant matrix replaces the column of the coefficients of the variable being solved for. Ex 13: Solve the system of equations using Cramer s Rule. x 7y 3 3x5y 17 3 7 3 5 7 17 17 5 15 119 104 x 4 1 7 1 5 7 3 5 21 26 3 5 1 3 1 17 3 3 3 17 17 9 26 y 1 1 7 26 26 26 3 5 Check: x 7y 3: 4 7 1 3 3 3 3x 5y 17 : 3 4 5 1 17 12 5 17 17 17 Solution: 4, 1 Solving a System of Three Equations and Three Unknowns with Cramer s Rule Ex 14: Solve the system using Cramer s Rule. x 3y z 1 2x 6y z 3 3x 5y 2z 4 Note: We will need the determinant of the coefficient matrix to find the value of each variable, so we will calculate this first. 1 3 1 1 3 1 1 3 2 6 1 2 6 1 2 6 12 9 10 12 5 18 31 35 4 3 5 2 3 5 2 3 5 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 26 of 49 4/3/2014

To find x, replace the x-column of the coefficient matrix with the constant matrix. 1 3 1 1 3 1 1 3 3 6 1 3 6 1 3 6 4 5 2 4 5 2 4 5 12 12 15 18 5 24 39 47 8 x 2 4 4 4 4 4 To find y, replace the y-column of the coefficient matrix with the constant matrix. 1 1 1 1 1 1 1 1 2 3 1 2 3 1 2 3 3 4 2 3 4 2 3 4 6 3 8 4 4 9 17 17 0 y 0 4 4 4 4 4 To find z, replace the z-column of the coefficient matrix with the constant matrix. 1 3 1 1 3 1 1 3 2 6 3 2 6 3 2 6 3 5 4 3 5 4 3 5 24 27 10 24 15 18 61 57 4 z 1 4 4 4 4 4 Solution: 2,0,1 (Check this on your own in the original three equations.) Note: We could also use Cramer s Rule to find x and y, then substitute these values into one of the original equations to find z. Graphing Calculator Evaluating Determinants Ex 15: Evaluate 6 12 5 det 10 3 1. 9 14 18 Step One: Entering in a Matrix In the Matrix Menu, choose Edit, then Matrix A. Our matrix is a 3 3, so enter in the dimensions. Then enter in the entries of the matrix. Keystrokes: Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 27 of 49 4/3/2014

Step Two: Evaluating the Determinant In the Home Screen, go to the Matrix Menu, then MATH, and choose det(. Then go back to the Matrix Menu and choose Matrix A under NAMES. Keystrokes: Note: You can perform all operations with matrices on the graphing calculator. You Try: 1. The Bermuda Triangle is an area located off the southeastern Atlantic coast of the United States, and is noted for reports of unexplained losses of ships, small boats, and aircraft. Find the area of the triangle on the map with coordinates (-80.226529, 25.789106), (-66.1057427, 18.4663188), and (-64.78138, 32.294887) Answer: The Bermuda Triangle has an area of 107.102 square degrees. 2. Use Cramer s Rule to solve the system 2x 5y 3z 10 3x y 4z 8. 5x 2 y 7z 12 3. Find the area of the triangle with vertices shown in the coordinate grid. Use your graphing calculator. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 28 of 49 4/3/2014

4. The salary for each of the stars of a new movie is $5 million, and the supporting actors each receive $1 million. The total amount spent for the salaries of the actors and actresses is $19 million. If the cast has 7 members, use Cramer s Rule to find the number of stars in the movie. Answer: 3 QOD: In order to use Cramer s Rule, what must be true about the determinant of the coefficient matrix, and why? Sample CCSD Common Exam Practice Question(s): 1. What is the determinant of 2 1 3 2? A. 7 B. 2 C. 1 D. 1 2. Cramer s Rule is used to solve the system of equations below. 4x 5 y z 11 3x 2 y 2z 5 2x 6 y 3z 8 Which determinant represents the denominator for the solution of z? A. B. C. 11 4 5 5 3 2 8 2 6 4 11 5 3 3 2 2 8 6 4 5 11 3 2 5 2 6 8 4 5 1 D. 3 2 2 2 6 3 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 29 of 49 4/3/2014

Inverse and Identity Matrices Big Idea: Recall that in real numbers, two numbers are inverses if their product is the identity, 1. Similarly, for matrices, the identity matrix is a square matrix that when multiplied by another matrix, equals that same matrix. If A is any n n matrix and I is the n n identity matrix, then A I A and I A A. Two n n matrices are inverses of each other if their product is the 1 identity matrix. If matrix A has an inverse symbolized by A 1 1, then A A A A I. Some matrices do not have an inverse. Use the determinant to determine whether a matrix has an inverse. If the value of the determinant of a matrix is 0, the matrix cannot have an inverse. A matrix equation can be solved by multiplying both sides of the equation by the inverse matrix. Objectives: N.VM.C.9 Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. N.VM.C.10 Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. A.REI.C.8 Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable. A.REI.C.9 Solve systems of equations. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). Skill: The Student will use the identity matrix. The Student will find inverse matrices with or without technology. The Student will solve matrix equations. Recall (let a be a real number): Identity of Multiplication = 1 a 11a a 1 0 Identity Matrix 2 2 = I 2 0 1 a b 1 0 a 1 b0 a 0 b1 a b c d 0 1 c 1 d 0 c 0 d 1 c d Now verify on your own that 1 0 a b a b 0 1 c d c d 1 0 0 Identity Matrix 3 3 = 0 1 0 I3 0 0 1 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 30 of 49 4/3/2014

Recall: The product of a number and its multiplicative inverse is the identity. 1 a a 1 The inverse of matrix A is denoted by 1 A, and 1 1 A A A A I. Caution: A! A 1 1 Ex 16: Verify that if A 3 1 4 2, then 1 1 1 A 2 2. 3 2 Find the product: 1 1 3 3 1 1 0 4 2 3 1 3 0 1 2 4 1 2 2 4 2 2 2 2 1 3 1 1 2 3 1 1 2 2 2 A A Now verify on your own that the product 1 0 1 A A 0 1. Finding the Inverse of a 2 2. a If A c b d, then 1 1 d b A ad bc c a Note: ad bc 0 Ex 17: Find the inverse of 5 4 A 4 4. A 1 1 4 4 1 4 4 1 1 54 4 4 4 5 4 4 5 1 1.25 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 31 of 49 4/3/2014

Solving a Matrix Equation Recall: To solve the equation ax b, we multiply both sides by the multiplicative inverse of a to isolate the variable. 1 ax 1 b x b a a a To solve the matrix equation AX B for X, where A and B are matrices, multiply both sides of the equation by the inverse of A. Ex: Solve the equation AX 1 1 1 A AX A B X A B B if 1 0 10 6 8 A and B 6 2 4 12 2. Step One: Write the equation. 1 0 10 6 8 X 6 2 4 12 2 Step Two: Find the inverse, 1 A. A 2 0 2 0 1 1 1 1 0 1 1 2 0 6 6 1 2 6 1 3 2 Step Three: Solve for X. Multiply both sides of the equation by the inverse of A. 1 1 A AX A B 1 0 1 0 1 0 10 6 8 1 X 1 3 6 2 3 4 12 2 2 2 110 04 16 012 18 02 10 6 8 X 1 1 1 310 4 36 12 38 2 32 24 25 2 2 2 Finding Inverse Matrices on the Graphing Calculator Ex 18: Find the inverse of the matrix 3 1 2 1 5 6. 7 2 6 Step One: Enter the matrix into Matrix A. Step Two: On the Home Screen, bring up Matrix A. Then use the key to find the inverse. To make the entries fractions, go to the MATH Menu and choose Frac. Keystrokes: Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 32 of 49 4/3/2014

Note: To view the answers, use the right arrow. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 33 of 49 4/3/2014

Solving a Two-Step Matrix Equation Ex 19: Solve the equation 7 9 3 4 1 9 X 4 5 4 3 6 6 for X. Step One: Isolate the X term by subtracting the matrix 3 4 4 3 from both sides. 7 9 3 4 3 4 1 9 3 4 4 5 X 4 3 4 3 6 6 4 3 7 9 2 5 X = 4 5 2 3 Step Two: Isolate X by multiplying both sides by the inverse of 7 9 4 5. Inverse of 7 9 4 5 7 5 9 4 4 7 35 36 4 7 4 7 1 5 9 1 5 9 5 9 = 5 9 7 9 5 9 2 5 4 7 4 5 X 4 7 2 3 5 2 9 2 55 9 3 8 2 X 4 2 7 2 4 5 7 3 6 1 You Try: Solve the matrix equation 7 5 8 2 9 3 X 4 3 6 1 6 2. QOD: What is the special relationship between a matrix and its inverse? Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 34 of 49 4/3/2014

1. Find the inverse of the matrix. Sample Questions: b. d. inverse does not exist ANS: C DOK 1 2. Find the inverse of the matrix, if it is defined. b. d. ANS: B DOK 1 3. Find the inverse of the matrix. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 35 of 49 4/3/2014

b. d. ANS: A DOK 1 4. Determine whether and are inverses. a. Yes, they are inverses. b. No, they are not inverses. ANS: B DOK: 2 5. Which of the following conditions would make AB = BA true for two square matrices? a. A and B must contain only positive integers. b. Both matrices must be or larger. c. Either A or B must be the identity matrix. d. B must be equal 2A. ANS: C DOK 1 6. Which multiplicative properties of equality hold for square matrices? a. Only the commutative and associative properties. b. Only the commutative and distributive properties. c. Only the associative and distributive properties. d. The commutative, associative, and distributive properties all hold for the multiplication of square matrices. ANS: C DOK 1 7. If J, K, and L are matrices and K = L -1, which statement must be true? Why? a. JKL = JLK because matrix multiplication is commutative. b. KJL = JLK because matrix multiplication is associative. c. JKL = KJL because matrix multiplication is commutative. d. KJL = LJK because matrix multiplication is associative. ANS: A DOK 3 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 36 of 49 4/3/2014

8. If, find Y such that WY = YW. a. b. c. d. ANS: B DOK 3 9. Use an inverse matrix to solve the linear system. Which of the following shows the correct solution? b. d. ANS: B DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 37 of 49 4/3/2014

10. The inverse of the coefficient matrix is given. Use the inverse to solve the linear system. b. d. ANS: A DOK 2 11. Tasty Bakery sells three kinds of muffins: chocolate chip muffins at 15 cents each, oatmeal muffins at 20 cents each, and cranberry muffins at 25 cents each. Charles buys some of each kind and chooses three times as many cranberry muffins as chocolate chip muffins. If he spends $3.70 on 17 muffins, how many cranberry muffins did he buy? a. 8 b. 9 c. 3 d. 5 ANS: B DOK 2 12. Choose the statement that is true about the given quantities. Column A x when solving the equation Column B y when solving the equation a. The quantity in column A is greater. b. The quantity in column B is greater. c. The two quantities are equal. d. The relationship cannot be determined from the given information. ANS: B DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 38 of 49 4/3/2014

13. Find the inverse of both A = and B = to determine. b. d. ANS: B DOK 2 14. Find the inverse of the matrix. ANS: DOK1 15. Solve the matrix equation. ANS: DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 39 of 49 4/3/2014

16. Jacob must simplify the expression (JK)(LM), where J, K, L, and M are matrices. If K is the multiplicative inverse of L and M is the multiplicative inverse of J, explain how Jacob can quickly show that the product is equal to the identity matrix,. ANS: DOK 4 The multiplication of square matrices is associative. Therefore, Jacob can rewrite (JK)(LM) as J(KL)M. Since K is the multiplicative inverse of L, and J(KL)M = JIM. Because JI = J, JIM can be rewritten as JM, and M is he multiplicative inverse of J,. 17. When and, MN = NM. Part A: What special relationship is shared by M and N that causes the two products to be equal? Part B: Is this true for all pairs of matrices sharing this relationship? Explain. ANS: DOK 4 Part A: M and N are multiplicative inverses. Both MN and NM are equal to the identity matrix,. Part B: In general, the Commutative Property of Multiplication does NOT hold for every pair of square matrices. However, the property will hold for all pairs of inverse matrices. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 40 of 49 4/3/2014

Using Matrices to Solve Systems of Equations Big Idea: Systems of Equations can be solved using an Inverse Matrix. Objectives: A.REI.C.8 Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable. A.REI.C.9 Solve systems of equations. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). Skill: The student will solve systems of equations using matrices. Solving Systems of Equations Using Inverse Matrices Ex: Solve the system 2 x 7 y 3 3x 8y 23 using an inverse matrix. Step One: Rewrite the system of equations as a matrix equation AX matrix, X is the variable matrix, and B is the matrix of constants. B, where A is the coefficient 2 7 x 3 3 8 y 23 Note: Use matrix multiplication to show that this represents the original system. Step Two: Find the inverse of Matrix A. A 2 7 3 8 1 1 8 7 1 8 7 A 2 8 73 3 2 37 3 2 Note: Do not multiply by the scalar. Step Three: Multiply both sides of the equation by 1 A. 1 8 7 2 7 x 1 8 7 3 37 3 2 3 8 y 37 3 2 23 x 1 8 3 7 23 1 185 5 y 37 3 3 2 23 37 37 1 Solution: 5,1 Therefore, x 5 y 1 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 41 of 49 4/3/2014

Check the solution in the original equations. 2x7y 3 2 5 7 1 3 10 7 3 3 3 3x8y 23 3 5 8 1 23 15 8 23 23 23 Now try the same problem on the graphing calculator: Enter the coefficient matrix into Matrix A and the constant matrix into matrix B. Find the product 1 A B on the home screen. Application Problem System of Three Equations Ex 20: Lawrence has $25 to spend on picking 21 pounds of three different types of apples in an orchard. Empire apples cost $1.40 per pound, Red Delicious apples cost $1.10 per pound, and Golden Delicious apples cost $1.30 per pound. He wants twice as many Red Delicious apples as the other two kinds combined. How many pounds of each type of apple should Lawrence pick? Let E = # of pounds of Empire apples, R = # of pounds of Red Delicious apples, and G = # of pounds of Golden Delicious apples Total Pounds: E R G 21 Total Cost: 1.4E 1.1R 1.3G 25 Other Information: R 2 E G 2E R 2G 0 Now we have a system of three equations with three variables. Rewrite the system as a matrix equation. 1 1 1 E 21 1.4 1.1 1.3 R 25 2 1 2 G 0 Enter the coefficient matrix as Matrix A and the constant matrix as Matrix B in the graphing calculator. On the Home Screen, multiply the Inverse of Matrix A by Matrix B. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 42 of 49 4/3/2014

Solution: Lawrence should pick 5 pounds of Empire apples, 14 pounds of Red Delicious apples, and 2 pounds of Golden Delicious apples. You Try: Solve the system the graphing calculator. x y 1 2x3y 12 using an inverse matrix (by hand). Check your answer using QOD: Explain why the solution of AX B is not X BA 1. Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 43 of 49 4/3/2014

Sample Questions Sample CCSD Common Exam Practice Question(s): 1. The flower shop in a grocery store sells flowers individually. The relationship between r, the cost of one rose, and c, the cost of one carnation, is represented by the matrix equation below. 3 2 r 8 2 5 c 9 What is the cost of buying one rose? A. $0.82 B. $1.00 C. $1.55 D. $2.00 2. Given the system of linear equations: 2x y 3 4x 5 y 0 Which equation below shows the solution to the system using inverse matrices? A. B. 1 x 2 1 3 y 4 5 0 1 x 3 2 1 y 0 4 5 x 2 1 y 4 5 C. 3 0 x 1 3 D. y 2 1 0 4 5 1 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 44 of 49 4/3/2014

Sample Questions: 1. Represent the system of equations as a matrix equation. b. d. ANS: C DOK 1 2. Represent the system of equations as a matrix equation. b. d. ANS: D DOK 1 3. Represent the system of equations as a matrix equation. b. d. ANS: D DOK 1 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 45 of 49 4/3/2014

4. Write the system of equations represented by the matrix equation. b. d. ANS: B DOK 2 5. Write the system of equations represented by the matrix equation. b. d. ANS: D DOK 2 6. Write the matrix equation for the system, and solve. a. b. c. d. ; (24, 106) ; (56, 26) ; (76, 18) ; (16, 6) Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 46 of 49 4/3/2014

ANS: B DOK 2 7. Solve the matrix equation for x and y. ANS: DOK 2 8. Solve the matrix equation for x and y. ANS: DOK 2 9. Use an inverse matrix to solve the linear system. ANS: (1, 2) DOK 2 10. Use an inverse matrix to solve the linear system. ANS: (4, 1) DOK 2 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 47 of 49 4/3/2014

ESSAY 1. Cajun Joe makes a gumbo that uses sausage, rice, and vegetables. The sausage costs $3 a pound, the rice costs $1.50 a pound, and the vegetables cost $0.75 a pound. Joe makes batches of gumbo that use a total of 20 pounds of ingredients and costs $1.50 per pound. Joe uses the same weight of rice as he does vegetables. Part A: Write a system of equations for this situation using x as the number of pounds of sausage used, y as the number of pounds of rice used, and z as the number of pounds of vegetables used. Then write a matrix equation for the system. Part B: Use an inverse matrix to solve the system. Part C: Joe also makes a "meaty" version of his gumbo that uses the same amount of sausage as rice in each 20 pound batch and costs $1.65 per pound. Write a system of equations for the meaty version of Joe's gumbo. Then use an inverse matrix to solve the system. Explain why the meaty version costs more per pound. ANS: Part A: ; Part B: x = 4 pounds of sausage, y = 8 pounds of rice, z = 8 pounds of vegetables Part C: x = 6 pounds of sausage, y = 6 pounds of rice, z = 8 pounds of vegetables The meaty version costs more because it uses more of the most expensive ingredient, sausage, and less of a less expensive ingredient, rice. DOK 3 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 48 of 49 4/3/2014

2. Heating fuel Company A charges $5 per gallon plus a $50 delivery fee. Heating fuel Company B charges $6 per gallon but gives a $25 discount per visit. Samantha wants to write a system of equations to represent the situation. Part A: Help Samantha by writing the system of equations that represent the situation. Include a description of what each variable represents. Part B: Write a matrix equation to represent the system. What did you need to do to the system before writing the matrix equation? Part C: In your matrix equation, is the order of the coefficient and variable matrices important? Justify your answer. Part D: Julia says her matrix equation is different than Samantha s. Two rows in her coefficient matrix are switched, and the same two rows in her constant matrix are switched. Did Julia make a mistake? Explain. ANS: Part A: where x is the amount of heating fuel in gallons and y is the cost in dollars, per visit. Part B: Sample answer: First rearrange the terms in each equation so that they are in the form. Part C: Yes, because order in matrix multiplication is important. In this case, you cannot multiply a matrix by a matrix because the inner dimensions are not the same. So, only the matrix product forms a new matrix, and is not defined. Part D: Sample answer: No, Julia s matrix equation is equivalent. This is because each row of coefficients and constants relate directly to the linear equation it represents, and the linear equations in a system can be written in either order as shown. DOK 4 Algebra 2 Unit 04 Notes Matrices & Determinantsrev Page 49 of 49 4/3/2014