CONDENSED LESSON 6. Matri Representations In this lesson, ou Represent closed sstems with transition diagrams and transition matrices Use matrices to organize information Sandra works at a da-care center. Last Monda and Wednesda afternoons, the children could choose to either finger paint or pla.8 games. Of the children who finger painted on Monda, 8% finger painted again on Wednesda, while % plaed games. Of the children who plaed games on Monda, 6% plaed games again on Wednesda, while % finger painted. Sandra made a diagram to displa this information. The arrows and labels show the patterns of the children s activities. For eample, the circular arrow labeled.6 indicates that 6% of the children who plaed games on Monda also plaed games on Wednesda. The arrow labeled. indicates that % of the children who plaed games on Monda finger painted on Wednesda. Diagrams like the one above are called transition diagrams because the show how something changes from one time to the net. You could show the same information in a transition matri. A matri is a rectangular arrangement of numbers. At right is a transition matri for Sandra s da-care information. Read the first three paragraphs of the lesson in our book, which present another eample of a transition diagram and a transition matri. Investigation: Chill Choices Complete the investigation in our book on our own, and then read the answers below. Step.9..9 Finger paint Monda.. Finger paint Pla games.8. Pla games Wednesda Finger Pla paint games..6 This entr shows that % of the children who plaed games on Monda finger painted on Wednesda..6 Ice cream. Frozen ogurt Step This Week Ice cream Yogurt Net Week Ice cream Yogurt.9...9 Step In the second week, 9% of the ice-cream eaters and % of the ogurt eaters choose ice cream. This means that (.9) (.) students choose ice cream. Using similar reasoning, (.) (.9) 9 students choose ogurt. Discovering Advanced Algebra Condensed Lessons CHAPTER 6 79
Lesson 6. Matri Representations Step In the second week, there are ice-cream eaters and 9 frozen-ogurt eaters. These calculations give the values for the third week: (.9) 9(.) ice-cream eaters (.) 9(.9) 7 frozen-ogurt eaters Step Let i n and f n be the number of ice-cream eaters and frozen-ogurt eaters in week n. Then the recursive routine is i and f i n i n (.9) f n (.) where n f n i n (.) f n (.9) where n Step 6 The number of ice-cream eaters will approach 6, and the number of frozen-ogurt eaters will approach 8. One wa to figure this out is to enter the recursive formulas for i n and f n into our calculator and calculate their values for large values of n. Matrices are useful for organizing information. Matri [B] at right represents the number of students b grade level at North and South High Schools. The rows, from top to bottom, represent the freshmen, sophomores, juniors, and seniors; the columns, from left to right, represent North High School and South High School. For eample, the indicates that there are juniors at North High School. The dimensions of a matri give the numbers of rows and columns. Matri [B] above has dimensions. Each number in a matri is called an entr, or element, and is identified as b ij, where i and j are the row and column numbers, respectivel. In matri [B], b, the entr in row, column. Eample A in our book shows how ou can use a matri to represent the vertices of a polgon. Read that eample carefull. Eample B returns to Karina s surve of skiers and snowboarders from the beginning of the lesson in our book. Read that eample carefull. To make sure ou understand the ideas, work through a similar series of steps to solve this problem: In Sandra s da-care center, children finger painted on Monda and students plaed games. How man children did each activit on Wednesda? Organize the information for Wednesda in a matri in the form [B] 7 89 7 9 [number of finger painters number of game plaers] Note that transition diagrams and matrices show change in a closed sstem that is, a sstem in which nothing is added or removed. Diagrams work well for representing relativel simple problems, but the can be difficult to use for situations in which there are man starting conditions. In such situations, a transition matri is usuall clearer and easier to interpret. 8 CHAPTER 6 Discovering Advanced Algebra Condensed Lessons
CONDENSED LESSON 6. Matri Operations In this lesson, ou Add and subtract matrices and multipl a matri b a scalar Use matri operations to transform a geometric figure Solve real-world problems that require ou to multipl matrices A matri is a compact wa of organizing data. Using matrices allows ou to perform operations on our data. In this lesson ou will see how this is useful. The tet on page 7 of our book illustrates how to add two matrices. In general, ou add (or subtract) two matrices b adding (or subtracting) the corresponding entries. You can add or subtract matrices onl if the have the same dimensions. This eample shows how to calculate the difference of two matrices. 9 7 7 9 7 7 Eample A in our book illustrates how ou can use matri operations to transform a triangle. Read this eample carefull. In general, ou translate a triangle horizontall h units and verticall k units b adding the matri 8 h k 7 h k h k 8 7 to the coordinate matri, and ou dilate a triangle b a factor of a b multipling the coordinate matri b the scalar, a. You can use these same methods to transform an polgon. You might want to draw our own polgon on a coordinate grid, represent it with a matri, and then eperiment with using matri operations to transform it. Multipling matrices is a bit more complicated than adding matrices or multipling a matri b a scalar. Eample B in our book uses a problem from Lesson 6. to show how to multipl a matri with onl one row b another matri. Read this eample ver carefull. You will get more practice with matri multiplication in the investigation and Eample C. 6 Investigation: Find Your Place Read the introduction and Step in the investigation in our book. The simulation is virtuall impossible to do b ourself, but make sure ou understand the instructions. If possible, ask one of our classmates about what happened during the simulation, and borrow his or her data so that ou can compare the theoretical results to what actuall happened. Complete the rest of the investigation on our own. You will not be able to answer Step. Assume cars start in Cit A, in Cit B, and in Cit C. When ou have finished the investigation, read the answers on the net page. Discovering Advanced Algebra Condensed Lessons CHAPTER 6 8
Lesson 6. Matri Operations Step This diagram displas the rules of the simulation. The diagram indicates, for eample, that, theoreticall, in each round, % of the cars in Cit C will move to Cit B, % will move to Cit A, and 7% will sta at Cit C. Here is the transition matri for the situation. Make sure ou understand what each entr represents...... C...7.7 Step Assume cars start in Cit A, in Cit B, and in Cit C. These initial conditions can be represented b the matri [ ]. The matrices are multiplied below. The entries of the product matri are the numbers of cars in Cities A, B, and C, respectivel, after the first transition. [..... [. ]...7 8. ] Think about how the calculations relate to the situation. For eample, in the first transition, % of the cars in Cit A sta in Cit A, % of the cars in Cit B move to Cit A, and % of the cars in Cit C move to Cit A. Therefore, the new number of cars in Cit A is (.) (.) (.). This is the sum of the products of the entries of the initial condition matri and the entries in the first column of the transition matri. How are the other entries of the product matri calculated? Step These matrices show the number of cars at each cit for each of the net four weeks. Week : [9.6 7..9] Week : [9.9 6.9.86] Week : [9. 6.69.] Week : [8.988 6.78.8] If ou continue multipling each result b the transition matri, ou will find that the long-run values are [9 6 ].. A.... B.. In the products ou have calculated so far, the matri on the left had onl one row. Eample C in our book shows how to find the product when the left matri has more than one row. Follow along with this eample, using a pencil and paper. Keep in mind that learning how to multipl matrices takes practice. As ou work on our homework eercises, ou will become more comfortable with the process. Eample C shows that ou can multipl two matrices onl if the number of columns in the left matri is the same as the number of rows in the right matri that is, if the inside dimensions of the matrices are the same. The outside dimensions tell ou the dimensions of the product matri. For eample, ou can multipl a 6 matri b a 6 matri because the inside dimensions are both 6. The result will be a matri. You cannot multipl a 6 matri b a 6 matri because the inside dimensions, and, are not the same. The tet in the Matri Operations bo on page in our book summarizes what ou learned in the lesson. Read this tet and then practice the operations until ou re comfortable with them. 8 CHAPTER 6 Discovering Advanced Algebra Condensed Lessons
CONDENSED LESSON 6. Row Reduction Method In this lesson, ou Write sstems of equations as augmented matrices Solve sstems of equations using the row reduction method Lesson 6. introduces a method for solving sstems of equations using matrices. Read the lesson in our book up through Eample A. The tet immediatel following Eample A shows how to represent row operations smbolicall. In the eample below, this notation is used to summarize each step. Tr to solve the sstem in the eample ourself before reading the solution. EXAMPLE Solution Use the row reduction method to solve this sstem. The equations are in standard form, so cop the coefficients and constants into an augmented matri. Perform row operations to transform this matri into the solution matri. 9 Add times row to row to get for m : R R R 9 Add times row to row to get for m : R R R 9 R Divide row b 7 to get for m : 7 R Add times row to row to get for m : R R R 7 8 The last column of the solution matri indicates that the solution is (, ). Investigation: League Pla Complete the investigation in our book. The results are given below. Note that the results use the data points (, ), (, ), and (, ) from Step. You ma want to use those points as well to make checking our work easier. Step At right is a scatter plot of the data. The graph is not linear. It appears to be quadratic. Step Using the points (, ), (, ), and (, ), ou get the following sstem of three equations. a b c a b c 6a b c Discovering Advanced Algebra Condensed Lessons CHAPTER 6 8 [,,,,, ]
Lesson 6. Row Reduction Method Step The augmented matri for the sstem in Step is Step Add times row to both row and row. In shorthand notation, this is R R R and R R R.The resulting matri is 6 6 Step Here is one possible series of steps: R 6 R and R 6 R 6 R R R and 6 R R R R R R R R R Step 6 R R and R R The solution appears in the last column: a, b, and c. So, the equation is. Step 7 Answers will var. Read the remainder of the lesson in our book. Be sure to tr to solve Eample B before reading the solution. 8 CHAPTER 6 Discovering Advanced Algebra Condensed Lessons
CONDENSED LESSON 6. Solving Sstems with Inverse Matrices In this lesson, ou Find an identit matri Find the inverse of a matri Use inverse matrices to solve equations In earlier math courses, ou learned that the number is the multiplicative identit. This means that when ou multipl an real number b, the number does not change. Similarl, when ou multipl a matri b an identit matri, the matri does not change. Read Eample A in our book, which shows ou how to find the identit matri for.the result,, is the identit for all matrices. Test this b choosing another matri and multipling it on either side b. In general, an identit matri is a square matri with s along the main diagonal, from top left to bottom right, and s for all the other entries. The shorthand notation [I] is often used to represent an identit matri. Identit matrices come in all (square) sizes. For eample, the identit matri for matrices is In earlier math courses, ou also learned that ever nonzero real number has a multiplicative inverse, the number ou multipl it b to get the multiplicative identit,. For eample, the multiplicative inverse of is because. Similarl, some (but not all) square matrices have an inverse matri. The inverse of a matri [A] is often denoted [A].When ou multipl a matri on either side b its inverse matri, ou get the identit matri. That is, [A][A] [I ] and [A] [A] [I]. In the investigation ou will find the inverse of a matri. Investigation: The Inverse Matri Follow Steps 7 of the investigation in our book. Then, compare our results with those below. Step A matri multiplied b its inverse matri is equal to the identit matri. So, a b c a c b d Step a c d b d Discovering Advanced Algebra Condensed Lessons CHAPTER 6 8
Lesson 6. Solving Sstems with Inverse Matrices Step a c b d a c b d Adding times a c to a c gives c. So, a, which means a.. Adding times b d to b d gives d. So, b, which means b.. So, a., b., c, and d. The inverse matri is therefore. Step Your calculator should confirm that [A]. Step.. (.) () (.) () and...().().().(). (.) () () () (.) () () () Both products are the same, but matri multiplication is not alwas commutative. For eample, let [B] and compare [A][B] to [B][A]. Step 6 Use our calculator to tr to find the inverses of the matrices. You will get an error message in ever case, which indicates that the matrices do not have inverses. A square matri does not have an inverse when one row is a multiple of the other... Step 7 Onl square matrices have inverses. Possible eplanation: Suppose a matri [B] with dimensions m n has an inverse. Then, [B][B] [I]. An identit matri must be square, so [I] must have dimensions m m. However, because it is also true that [B] [B] [I], [I] must have dimensions n n. Therefore, m must equal n, so [B] is square. Recall that ou can solve an equation in the form a b b multipling both sides b the multiplicative inverse of a. For eample, to solve, multipl both sides b to get.in a similar wa, if ou rewrite a sstem of equations in matri form, ou can solve it b multipling both sides b the inverse of the coefficient matri. Read the Solving a Sstem Using the Inverse Matri bo in our book and then read Eample B ver carefull, following along with a pencil and paper. To make sure ou understand the method, tr the following eample. EXAMPLE Solve this sstem using an inverse matri. 86 CHAPTER 6 Discovering Advanced Algebra Condensed Lessons
Lesson 6. Solving Sstems with Inverse Matrices Solution The matri for this sstem is Use a calculator to find that Multipl both sides of the equation b the inverse matri on the left side......... So, the solution to the sstem is (,.). Now, read the remainder of the lesson, including Eample C, which involves solving a sstem of three equations. Discovering Advanced Algebra Condensed Lessons CHAPTER 6 87
CONDENSED LESSON 6. Sstems of Linear Inequalities In this lesson, ou Write sstems of linear inequalities to describe real-world situations Graph the solution, or feasible region, for a sstem of inequalities Find the vertices of a feasible region Real-world situations involving a range of values can often be represented b inequalities. The table at the beginning of Lesson 6. in our book gives several eamples. You can perform operations on both sides of an inequalit, just as ou can for equations. However, when ou multipl or divide both sides of an inequalit b a negative quantit, ou must reverse the inequalit smbol. Investigation: Paing for College Read the first paragraph of the investigation in our book, and then complete Steps. When ou are finished, compare our results to those below. Step,,,, Step Here are some possible (, ) pairs: (, ), (, ), (, ), (, 9999), (, ). The sum can be less than $, because the administrators don t have to invest all the mone. Step The solutions to are below the line., Step Points for which one or both of the coordinates are negative, for eample, (, ) or (, 6), do not make sense in this situation., Step Read the paragraph before Step. Step asks ou to translate all of the limitations, or constraints, in that paragraph into a sstem of inequalities. Tr this ourself, and then compare our sstem to the one below.,, The amount invested in stocks must be at least $. The amount invested in bonds must be at least $. The amount invested in stocks must be at least $. The amount invested in bonds must be at least $. The amount invested in bonds is at least twice the amount invested in stocks. Total amount invested must be less than or equal to $,. Discovering Advanced Algebra Condensed Lessons CHAPTER 6 89
Lesson 6. Sstems of Linear Inequalities Step 6 The solutions to this sstem are the values that satisf all the inequalities. You can t list all the solutions (there are an infinite number of them), but ou can show the region on a graph. To do this, graph the solution of each of the inequalities. The solution to the sstem is the area where all the graphs overlap. In this case, the solution is all the points on or to the right of and on or above and on or above, and on or below. The solution is shown at right. C The solution to an inequalit is called a feasible region. B To find the corners, or vertices, of the region, ou need to find,, the points where the lines forming each corner intersect. This A involves solving these sstems:,, The solutions to these sstems are (, ),, 6666, and (, ). So, ou can describe the feasible region for the sstem as the triangle with vertices (, ),, 6666, and (, ), and its interior. The tet between the investigation and the eample in our book summarizes the work ou did in the investigation. Read that tet and then work through the eample. Below is another eample. EXAMPLE Solution Sketch the feasible region for this sstem of inequalities, and identif its vertices.. Here are the graphs of each inequalit:. The feasible reason is the overlap of the graphs as shown at right. You can read the vertices from the graph or find them b solving these sstems:.. The solutions are (,.), (, ), and (, ). Read the remainder of the lesson in our book. 9 CHAPTER 6 Discovering Advanced Algebra Condensed Lessons
CONDENSED LESSON 6.6 Linear Programming In this lesson, ou Use the method of linear programming to solve problems that involve maimizing or minimizing the value of an epression Linear programming is the process of finding a feasible region and then finding the point within the region that gives the maimum or minimum value for a specific epression. Read about linear programming in the first three paragraphs of the lesson in our book. Investigation: Maimizing Profit Step Read the first paragraph of the investigation in our book. Below, the given information is organized into a table. In addition to this information, note that the number of unglazed birdbaths,, must be greater than or equal to 6. Amount per Amount per Constraining unglazed birdbath glazed birdbath value Wheel hours. 8 Kiln hours 8 6 Profit $ $ Maimize Step Use our table to write inequalities that reflect the given constraints, along with an commonsense constraints. Then, compare our inequalities with those below.. 8 8 6 6 Potter-wheel hours constraint Kiln hours constraint Constraint on number of unglazed birdbaths Common sense Common sense Now, make a graph of the feasible region for the sstem of inequalities, and label the vertices. Compare our graph to the graph at right. Step It makes sense to produce onl whole numbers of birdbaths. List the coordinates of all the points within the feasible region for which both coordinates are whole numbers. Be sure to include points on the boundar lines. Your list should include these points: (6, ), (7, ), (8, ), (9, ), (, ), (, ), (, ), (, ), (, ), (, ), (6, ), (6, ), (7, ), (8, ), (9, ), (, ), (, ), (, ), (, ), (, ), (6, ), (7, ), (8, ) These points represent all the possible combinations of unglazed and glazed birdbaths the shop can produce. Discovering Advanced Algebra Condensed Lessons CHAPTER 6 9
Lesson 6.6 Linear Programming Step The shop makes $ per unglazed birdbath and $ per glazed birdbath. The equation for the profit, P, if the compan produces unglazed birdbaths and glazed birdbaths is P. Find the profit for each of the feasible points from Step. You should get the results below. Point Profit (6, ) $6 (9, ) $9 (, ) $ (, ) $ (7, ) $ (, ) $ (, ) $7 (7, ) $ Point Profit (7, ) $7 (, ) $ (, ) $ (6, ) $6 (8, ) $ (, ) $ (, ) $8 (8, ) $6 Point Profit (8, ) $8 (, ) $ (, ) $ (6, ) $ (9, ) $ (, ) $6 (6, ) $ Step The shop will make a maimum profit of $8 if it produces unglazed birdbaths and glazed birdbath. The point (, ) is a verte of the feasible region. Complete Steps 6 8, and then compare our results to those below. Step 6 Step 7 Step 8 ; see graph at right. ; see graph at right. 7; see graph at right. Step 9 Look at the profit lines from Steps 6 8. Notice that the are all parallel and that as the profit increases, the lines move up and to the right. If ou imagine continuing to move the profit line up and to the right, alwas keeping the same slope, the last point in the feasible reason it will pass through is (, ). Therefore, (, ) must be the point that maimizes profit. This same method would work with other situations as well. If the verte did not have integer coordinates, ou could test the integer points near the verte. To minimize the profit, ou could imagine sliding the profit line down and to the left. The last point in the feasible reason it would pass through is (, ), which is the point that gives the minimum profit. The eample in our book provides an eample in which linear programming is used to minimize costs. Work through this eample and read the remainder of the lesson. The lesson ends with a bo summarizing the steps for solving a linear programming problem. 9 CHAPTER 6 Discovering Advanced Algebra Condensed Lessons