Matrix Operations and Equations
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1 C H A P T ER Matrix Operations and Equations 200 Carnegie Learning, Inc. Shoe stores stock various sizes and widths of each style to accommodate buyers with different shaped feet. You will use matrix operations to determine the inventory levels of shoes across a chain of stores.. Arrays, Arrays! Introduction to Matrices and Matrix Operations p..4 Inverses, Anyone? Row Operations and Multiplicative Inverses p School Daze Matrix Multiplication p..5 Solutions Abound! Matrix Equations p. 5. Commune, Associate? Properties of Matrix Operations p. 9.6 Determining Determinants! Determinants p. 45 Chapter l Matrix Operations and Equations
2 Mathematical Representations INTRODUCTION Mathematics is a human invention, developed as people encountered problems that they could not solve. For instance, when people first began to accumulate possessions, they needed to answer questions such as: How many? How many more? How many less? People responded by developing the concepts of numbers and counting. Mathematics made a huge leap when people began using symbols to represent numbers. The first numerals were probably tally marks used to count weapons, livestock, or food. As society grew more complex, people needed to answer questions such as: Who has more? How much does each person get? If there are 5 members in my family, 6 in your family, and 0 in another family, how can each person receive the same amount? During this course, we will solve problems and work with many different representations of mathematical concepts, ideas, and processes to better understand our world. The following processes can help you solve problems. Discuss to Understand Read the problem carefully. What is the context of the problem? Do you understand it? What is the question that you are being asked? Does it make sense? Think for Yourself Do I need any additional information to answer the question? Is this problem similar to some other problem that I know? How can I represent the problem using a picture, a diagram, symbols, or some other representation? Work with Your Partner How did you do the problem? Show me your representation. What else do we need to solve the problem? Does our reasoning and our answer make sense to one another? Work with Your Group Show me your representation. What else do we need to solve the problem? Does our reasoning and our answer make sense to one another? How can we explain our solution to one another? To the class? Share with the Class Here is our solution and how we solved it. We could only get this far with our solution. How can we finish? Could we have used a different strategy to solve the problem? This is the way I thought about the problem how did you think about it? This is the way I thought about the problem how did you think about it? 2 Chapter l Matrix Operations and Equations 200 Carnegie Learning, Inc.
3 . Arrays, Arrays! Introduction to Matrices and Matrix Operations Objectives In this lesson you will: l Write matrices. l Add and subtract matrices. l Multiply a matrix by a scalar. Key Terms l matrix l dimensions of a matrix l additive identity matrix l additive inverse matrix l scalar multiplication Problem What is a Matrix? 200 Carnegie Learning, Inc. A matrix is an array of numbers arranged in rows and columns. The plural of matrix is matrices. The dimensions of a matrix are the number of rows and columns. An n m matrix has n rows and m columns. The element a ij is the number in the ith row and jth column. a a 2 a n a 2 a 22 a n2 a m a 2m a nm Matrices are useful when arrays of numbers are needed. For example, consider a shoe store that must track the number of shoes of each size in stock. The shoes come in full sizes 9 through and in widths C, D, E, EE, and EEEE. The inventory of each size and width for a particular style of shoe can be organized using the following matrix. C D E EE EEEE Use this matrix to answer the following questions. a. What does each row represent? Each column? Lesson. l Introduction to Matrices and Matrix Operations
4 b. What are the dimensions of the matrix? Explain. c. What is a 4, the number in the rd row and 4th column? a 55? a 2? d. What does a 4 represent? a 2? 2. A matrix is named using a capital letter. The matrix in Question can be named matrix A. The inventory of the same shoes at another shoe store can be represented by matrix B B a. What does each row represent? Each column? b. What are the dimensions of matrix B? Explain. 200 Carnegie Learning, Inc. 4 Chapter l Matrix Operations and Equations
5 c. What is b 55? b 2? d. What does b 4 represent? b 2? Problem 2 Operations with Matrices. How can you determine the number of shoes of each size at both stores? What operation can you use? 2. Calculate A B.. Calculate A B. 200 Carnegie Learning, Inc. 4. Do the elements in the matrices in Questions 2 and make sense in terms of the problem situation? Lesson. l Introduction to Matrices and Matrix Operations 5
6 To add matrix A and matrix B, A B, add each element of matrix A, a ij, to the corresponding element of matrix B, b ij. To subtract matrix B from matrix A, A B, subtract each element of matrix B, b ij, from the corresponding element of matrix A, a ij. 5. Can any two matrices be added? Can any two matrices be subtracted? Explain. 6. Determine if it is possible to calculate each sum or difference. If so, calculate each sum or difference. If not, explain why. A B C D 2 8 E 2 a. A D b. B C c. C D d. A B 200 Carnegie Learning, Inc. e. D A 6 Chapter l Matrix Operations and Equations
7 f. (A D) E g. A (D E) h. A A. When are matrix addition and subtraction defined? 8. Is matrix addition commutative? Associative? Explain. 9. What matrix can be added to matrix A so that their sum is matrix A? 200 Carnegie Learning, Inc. An additive identity matrix is a matrix that when added to matrix A results in matrix A. 0. Is the additive identity matrix the same for every matrix? Explain.. What matrix can be added to matrix D so that the sum is the additive identity matrix? Lesson. l Introduction to Matrices and Matrix Operations
8 The additive inverse matrix of matrix A is a matrix that when added to matrix A results in the additive identity matrix. 2. For a real number x, x x 2x. For a matrix A, does A A 2A? Explain.. How can you calculate A A without adding each pair of elements? Scalar multiplication by a constant is multiplying each element of a matrix by a constant. 4. Calculate each resulting matrix. a. A b. A 2D c. E 2A d. (B C) e. B C 200 Carnegie Learning, Inc. 8 Chapter l Matrix Operations and Equations
9 Problem Matrices and Technology A graphing calculator can be used to perform matrix operations. To enter a matrix into a graphing calculator, perform the following steps: l Press the 2ND button and the x button to open the matrix menu. l Select EDIT and the name of the matrix. l Enter the dimensions of the matrix. Remember that an n m matrix has n rows and m columns. l Enter the elements of the matrix.. Input each matrix into a graphing calculator. A B C D 2 To use a matrix in an expression, perform the following steps: 8 E l Press the 2ND button and the x button to open the matrix menu. l Select NAMES and the name of the matrix. Press ENTER. l The name of the matrix will be inserted in the expression Determine whether each resulting matrix can be calculated. If so, calculate each resulting matrix using a graphing calculator. If not, explain why not. a. A D b. B C c. C D d. A B 200 Carnegie Learning, Inc. e. D A f. (A D) E g. A (D E) h. A A Lesson. l Introduction to Matrices and Matrix Operations 9
10 i. A j. A 2D k. E 2A l. (B C) m. B C. Compare your answers in Questions 2(i) through 2(m) to your answers in Problem 2, Questions 4(a) through 4(e). What do you notice? 4. What are the advantages and disadvantages of using a calculator to perform operations with matrices? 200 Carnegie Learning, Inc. Be prepared to share your solutions and methods with another pair, group, or the entire class. 0 Chapter l Matrix Operations and Equations
11 .2 School Daze Matrix Multiplication Objectives In this lesson you will: l Write row and column matrices. l Multiply matrices. l Transpose matrices. Key Terms l row matrix l column matrix l square matrix l transpose matrix Problem High School Daze In Masontown High School there are: l 25 ninth graders, of whom 6 are female; l 298 tenth graders, of whom 6 are female; l 256 eleventh graders, of whom 46 are female; l 226 twelfth graders, of whom 26 are female. On average, 8% of the female students and 85% of the male students attend school each day.. On average, how many ninth graders are in school each day? 200 Carnegie Learning, Inc. 2. On average, how many eleventh graders are in school each day? Lesson.2 l Matrix Multiplication
12 The truant officer reports that, on average, 2% of the female students and.5% of the male students are truant each day.. On average, how many tenth graders are truant each day? 4. On average, how many twelfth graders are truant each day? The guidance counselor reports that, on average, 2% of the female students and 9% of the male students are on the honor roll each grading period. 5. On average, how many ninth graders are on the honor roll each grading period? 6. On average, how many twelfth graders are on the honor roll each grading period?. How did you calculate the answers to Questions through 6? A row matrix is a matrix with one row. A column matrix is a matrix with one column. 8. Write a row matrix to represent the percentage of female students and male students who are in school each day. 200 Carnegie Learning, Inc. 9. Write a column matrix to represent the number of female students and male students in the ninth grade. 2 Chapter l Matrix Operations and Equations
13 0. Explain how to calculate how many ninth graders are in school each day using the row matrix from Question 8 and the column matrix from Question 9.. Write a column matrix to represent the number of female students and male students in the tenth grade. 2. Calculate the approximate number of tenth graders who are in school each day using the row matrix from Question 8 and the column matrix from Question.. Write a 2 4 matrix S to represent the number of males and females in each grade. Let the columns represent the grade level and the rows represent females and males. 200 Carnegie Learning, Inc. 4. Calculate the number of students at each grade level who are in school each day using the row matrix from Question 8 and matrix S from Question. Write the result in a row matrix. The process that you used to calculate the row matrix is one example of multiplying matrices. Lesson.2 l Matrix Multiplication
14 Problem 2 Multiplying Matrices Recall that to add or subtract two matrices, the matrices must have the same dimensions. To multiply two matrices, the number of columns in the first matrix must be the same as the number of rows in the second matrix. The product of an i j matrix and a j k matrix is an i k matrix. Matrix A and matrix B can be multiplied as follows: A 2 AB B (2) 4() ()() (2) (2)() 4() C (4) 4() ()(5) (4) (2)() 4(5) The element in the first row and first column of the product matrix is the sum of: l The product of a and b l The product of a 2 and b 2 l The product of a and b The element in the first row and second column of the product matrix is the sum of: l The product of a and b 2 l The product of a 2 and b 22 l The product of a and b 2 The element in the second row and first column of the product matrix is the sum of: l The product of a 2 and b l The product of a 22 and b 2 l The product of a 2 and b The element in the second row and second column of the product matrix is the sum of: l The product of a 2 and b Carnegie Learning, Inc. l The product of a 22 and b 22 l The product of a 2 and b 2 4 Chapter l Matrix Operations and Equations
15 . Calculate each product. a. CA b. BC c. BA 200 Carnegie Learning, Inc. 2. Enter Matrices A, B, and C into a graphing calculator. Then check the products you calculated in Question. Are the products the same?. Is it possible to calculate AC? Explain. 4. Is AB equal to BA? Explain. Lesson.2 l Matrix Multiplication 5
16 5. Is matrix multiplication commutative? Explain. 6. Determine whether each product can be calculated. If so, calculate each product. If not, explain why not. a. A 2 b. C 2 c. C. When is it possible to calculate powers of matrices? A square matrix is a matrix with the same number of rows and columns. 8. Let D and E 0 2. Calculate each product. 2 a. DE b. ED 9. Calculate each product, and then check your result with a graphing calculator. 200 Carnegie Learning, Inc. a. DA b. EA 6 Chapter l Matrix Operations and Equations
17 c. CE d. BE e. D 2 C Problem School Daze Revisited Let us return to Masontown High School. Recall that: l 25 ninth graders, of whom 6 are female; l 298 tenth graders, of whom 6 are female; l 256 eleventh graders, of whom 46 are female; l 226 twelfth graders, of whom 26 are female. l On average, 8% of the female students and 85% of the male students attend school each day. l The truant officer reports that, on average, 2% of the female students and.5% of the male students are truant each day. l The guidance counselor reports that, on average, 2% of the female students and 9% of the male students are on the honor roll each grading period. 200 Carnegie Learning, Inc.. Write a 2 matrix R to represent the percentage of females and males who attend school, are truant, and make the honor roll. Let the columns represent females and males. Let the rows represent school attendance, truancy, and honor roll. 2. Calculate the product of matrix R and matrix S from Problem, Question. Name the product matrix T. Lesson.2 l Matrix Multiplication
18 . What does each element of matrix T mean in terms of the problem situation? A transpose matrix is a matrix formed by writing the rows of the original matrix as columns and the columns of the original matrix as rows. Let matrix A Determine R T The transpose matrix of matrix A, written as AT, is A T 5. What does each row and column of R T represent? Be prepared to share your methods and solutions. 200 Carnegie Learning, Inc. 8 Chapter l Matrix Operations and Equations
19 . Commune, Associate? Properties of Matrix Operations Objectives In this lesson you will: l Identify the additive identity matrix. l Identify the additive inverse matrix. l Identify the multiplicative identity matrix. l Determine properties of matrix addition and multiplication. Key Term l multiplicative identity matrix Problem Addition of Matrices. What must be true about the dimensions of two matrices to calculate the sum? 2. Let G and H a. Calculate G H. b. Calculate H G. 200 Carnegie Learning, Inc.. Is matrix addition commutative? Explain. Lesson. l Properties of Matrix Operations 9
20 4. Let A a a 2 a n a 2 a 22 a n2 a m a 2m and B a nm b b 2 b n b 2 b 22 b n2 b m b 2m b nm Use matrix A and matrix B to prove that matrix addition is commutative.. 5. Is matrix addition associative? Explain. Recall that the additive identity matrix is defined as I such that for any matrix A, A I A. 6. Describe the elements of the additive identity matrix with respect to any matrix A. Recall that the additive inverse matrix is defined as B such that for any matrix A, A B I.. Describe the elements of the additive inverse matrix with respect to any matrix A. 200 Carnegie Learning, Inc. 20 Chapter l Matrix Operations and Equations
21 Problem 2 Multiplication of Matrices. What must be true about the dimensions of two matrices to calculate the product? 5 2. Let G 2 5 a. Calculate GK. 2 and K 2 5. b. Calculate KG.. Does it appear that matrix multiplication is commutative? Explain Let A a. Calculate AB. 2 and B Carnegie Learning, Inc. b. Calculate BA. 5. Does it appear that matrix multiplication is commutative for square matrices? Explain. Lesson. l Properties of Matrix Operations 2
22 6. Let A 2 a. Calculate (CA)B. 4 2, B 0 2 2, and C b. Calculate C(AB).. Does it appear that matrix multiplication is associative? Explain. 8. Let S 2 4, R a. Calculate S(R T ). 6 4, and T b. Calculate SR ST. 9. Does it appear that matrix multiplication distributes over matrix addition when the sums and products are defined? Explain. 200 Carnegie Learning, Inc. 22 Chapter l Matrix Operations and Equations
23 0. Let P and Q a. Calculate (P Q) b. Calculate P 2. c. Calculate Q 2. d. Calculate 2PQ. 200 Carnegie Learning, Inc. e. Calculate P 2 2PQ Q 2.. Is (P Q) 2 P 2 2PQ Q 2? Explain. Lesson. l Properties of Matrix Operations 2
24 The multiplicative identity matrix is defined as I such that for any matrix A, AI A. 2. Determine the elements of each multiplicative identity matrix. a b. 2 2 c If A is an n m matrix, then what are the dimensions of the multiplicative identity matrix I? 6 4. Describe the elements of the multiplicative identity matrix with respect to any matrix A. Be prepared to share your methods and solutions. 200 Carnegie Learning, Inc. 24 Chapter l Matrix Operations and Equations
25 .4 Inverses, Anyone? Row Operations and Multiplicative Inverses Objectives In this lesson you will: l Perform the three elementary row operations for matrices. l Use row operations to solve systems of linear equations. l Identify the multiplicative inverse of a matrix. l Use row operations to derive the multiplicative inverse of a matrix. Key Terms l elementary row operation for matrices l multiplying a row by a nonzero number l row addition l switching rows l row equivalent matrix l Gaussian elimination l multiplicative inverse of a matrix Problem Elementary Row Operations Matrices can be used to solve systems of linear equations. Before doing so, it is important to introduce the tools that can be used. There are three elementary row operations for matrices: l Multiplying a row by a nonzero number 200 Carnegie Learning, Inc. Any row, R, can be replaced by the product of that row and a nonzero number The product of the number 2 and row replaces row. l Row addition 2 4 Any row, R, can be added to or subtracted from any other row with the result replacing either row The sum of row and row 2 replaces row R replaces row R R 2 replaces row R 2 R replaces row The difference of row 2 and row replaces row Lesson.4 l Row Operations and Multiplicative Inverses 25
26 l Switching rows Any row can switch places with any other row R and R 2 are switched Row one and row two are switched. 6 4 A row equivalent matrix is a matrix that is formed by performing an elementary row operation on a matrix. Problem 2 Systems of Linear Equations and Matrices Previously, you solved systems of linear equations graphically, using substitution or elimination, and using determinants. These methods are useful for systems of linear equations in two variables, but can be cumbersome when solving systems of linear equations in three or more variables. Gaussian elimination, named after German mathematician Carl Friedrich Gauss, is a method of solving a system of linear equations using elementary row operations. To solve a system of linear equations using Gaussian elimination, perform the following steps. l Write each equation in the form ax by c. l Write the system of linear equations as a matrix. Each row represents one equation. The number of rows is equal to the number of equations. Each column represents the coefficients of one variable. The last column represents the constants. The number of columns is one more than the number of variables. l Perform elementary row operations to create row equivalent matrices. The final equivalent matrix includes only elements of and 0 in all but the last column with each column containing a in a different row. l The solution is represented by the numbers in the last column. 200 Carnegie Learning, Inc. 26 Chapter l Matrix Operations and Equations
27 Consider the system: 2x y 5 x 2y This system of linear equations is solved using elimination on the left and using Gaussian elimination on the right. 2x y 5 x 2y 2 4x 2y 0 x 2y 4 x Multiply R by 2. Replace R with the product 2R. 0 Add R and R 2. Replace R with the sum R R 2. 2 Divide R by. Replace R with the quotient R. x 2 x 2() y y 5 0 Solution: x, y Simplify the fractions. x. Multiply R by and subtract the product from R 2. Replace R 2 with the difference R 2 R. 2 Divide R by 2. Replace R with the quotient R y Check: () 2() 9 2 Check: 2() Carnegie Learning, Inc. Lesson.4 l Row Operations and Multiplicative Inverses 2
28 . Solve each system of linear equations using Gaussian elimination. Describe each step. x 2y a. 2x y 5 b. x 4y x 5y 200 Carnegie Learning, Inc. 28 Chapter l Matrix Operations and Equations
29 2. The multiplicative identity matrix is shown. Perform the same steps as you did in Question (b) The matrix representing the system of equations in Question (b) is shown with the last column removed. Multiply this matrix by the resulting matrix in Question 2. What do you notice? Carnegie Learning, Inc. Lesson.4 l Row Operations and Multiplicative Inverses 29
30 Problem Multiplicative Inverse of Square Matrices The multiplicative inverse matrix is defined as A such that for any matrix A, AA l where I is the identity matrix. You can calculate the multiplicative inverse of matrix A by transforming the equation AA I into the equation IA A using elementary row operations as shown. Let A 2 AA I A 0 A 0 A Check: 2 2 A 0 A 2 2 A Divide R by. Replace R with the quotient R Multiply R by 2 and add the product and R 2. Replace R 2 with the sum 2R R 2. 0 Multiply R 2 by Replace R 2 0 with the product R 2. Multiply R 2 by and add the product and R. Replace R with the sum R 2 R. 200 Carnegie Learning, Inc. 0 Chapter l Matrix Operations and Equations
31 . Determine the multiplicative inverse of B 2 your answer. 4, and then check Determine the multiplicative inverse of C 2 your answer. 4, and then check 200 Carnegie Learning, Inc. Lesson.4 l Row Operations and Multiplicative Inverses
32 . Show that the inverse of D a c b d is D ad bc d c b a. 200 Carnegie Learning, Inc. 2 Chapter l Matrix Operations and Equations
33 Problem 4 Multiplicative Inverse of Square Matrices and Technology To calculate the multiplicative inverse of a matrix using a graphing calculator, perform the following steps: l Input the matrix into the calculator. l Press the 2ND button and the x button to open the matrix menu. l Select NAMES and the name of the matrix. Press ENTER. l Press the x button. Press ENTER.. Determine the multiplicative inverse of each matrix using a graphing calculator. a. A 2 b. B c. C Carnegie Learning, Inc. Be prepared to share your methods and solutions. Lesson.4 l Row Operations and Multiplicative Inverses
34 200 Carnegie Learning, Inc. 4 Chapter l Matrix Operations and Equations
35 .5 Solutions Abound! Matrix Equations Objectives In this lesson you will: l Solve systems of linear equations in two variables using matrices. l Determine the equation of a line passing through two points using matrices. l Solve systems of linear equations in three variables using matrices. l Determine the equation of a parabola passing through three points using matrices. Key Term l matrix equation Problem Solving Matrix Equations in Two Variables A matrix equation is an equation consisting of matrices. 200 Carnegie Learning, Inc. Previously, you wrote a system of linear equations as a single matrix. A system of linear equations can also be written as a matrix equation. For example, the system of equations x y 2x y 4 can be written as: 2 x y 4 or A y x 4 To solve the matrix equation A y x, multiply both sides of the 4 equation by the multiplicative inverse A. The product of a matrix and its multiplicative inverse is the multiplicative identity matrix. So, this will isolate the matrix that includes just the variables. Lesson.5 l Matrix Equations 5
36 x y can be solved as follows. 4 The matrix equation 2 The multiplicative inverse of matrix A is A 2 A A x y A x y 8 5 x 5, y 8 y x 2 4 Check: x y (5) 8 2x y 2(5) (8) 4 Remember that there are three ways to calculate the inverse matrix. Method : Using a graphing calculator: l Input the matrix into the calculator. l Press the 2ND button and the x button to open the matrix menu. l Select NAMES and the name of the matrix. Press ENTER. l Press the x button. Press ENTER. Method 2: Determining the inverse by hand as in Lesson.4. Let A 2 AA I A 0 A 0 A A 0 A 2 0 Divide R by. Replace R with the quotient R. 0 Multiply R by 2 and add the product and R 2. Replace R 2 with the sum 2R R 2. 0 Multiply R by 2. Replace R with the product 2 R A 2 Multiply R 2 by and add the product and R. Replace R with the sum R R Carnegie Learning, Inc. 6 Chapter l Matrix Operations and Equations
37 Check: Then, multiply A Method : Use the formula for calculating the inverse of a 2 by 2 matrix: 0 If A a b, then A c d d b ad bc c a. Write each system of linear equations as a matrix equation, and then solve the matrix equation. x 4y a. x 5y 200 Carnegie Learning, Inc. b. 4x y x 5y 2 Lesson.5 l Matrix Equations
38 x 2y 2 c. x 4y 24 d. x 2.4y 29.2x 5.y Carnegie Learning, Inc. 8 Chapter l Matrix Operations and Equations
39 2. Determine the equation of a line passing through the points (2, 5) and (4, 4) using a matrix equation. a. Substitute each point into the slope-intercept form of a line y mx b. This results in two equations with the variables m and b. b. Write the system of equations as a matrix equation. c. Solve the matrix equation for the slope, m, and the y-intercept, b. d. Write the equation of the line.. Determine the equation of a line passing through the points (, 4) and (8, 6) using a matrix equation. 200 Carnegie Learning, Inc. Lesson.5 l Matrix Equations 9
40 Problem 2 Matrix Equations in Three Variables The same methods you used to solve a system of linear equations in two variables can also be used to solve systems of linear equations with three or more variables. As the number of rows and columns increase, it becomes more challenging to use Gaussian elimination. Calculating the multiplicative inverse is also more challenging. For instance, the formula for the inverse of a x matrix is: Let M a b c d e f g h i ci fh ch bi then M fg di ah cg a(ei fh) b(di fg) c(dh eg) dh eg bg ah bf ce cd af ae bd A graphing calculator can solve these systems quickly. If a variable is not seen in an equation, then place a zero in its location. 2x y 4z 6. Solve the system 2y z x 4z 9 a. Write the system of equations as a matrix equation. b. Calculate the multiplicative inverse. c. Solve the matrix equation. 200 Carnegie Learning, Inc. 40 Chapter l Matrix Operations and Equations
41 2x y z 9 2. Solve the system 6x 2y z 0 4x y 2z 0 a. Write the system of equations as a matrix equation. b. Calculate the multiplicative inverse. c. Solve the matrix equation. 200 Carnegie Learning, Inc.. Determine the equation of a parabola that passes through the points (, ), (2, 8), and (, ). The general form of the quadratic function whose graph is a parabola is y ax 2 bx c. a. Substitute each point into the equation y ax 2 bx c. This results in three equations with the variables a, b, and c. Lesson.5 l Matrix Equations 4
42 b. Write the system of equations as a matrix equation. c. Calculate the multiplicative inverse. d. Solve the matrix equation. 200 Carnegie Learning, Inc. 42 Chapter l Matrix Operations and Equations
43 4. Determine the equation of a parabola that passes through the points (, ), (, 9), and (.5, 9). Be prepared to share your methods and solutions. 200 Carnegie Learning, Inc. Lesson.5 l Matrix Equations 4
44 200 Carnegie Learning, Inc. 44 Chapter l Matrix Operations and Equations
45 .6 Determining Determinants! Determinants Objectives In this lesson you will: l Write and evaluate determinants. l Use Cramer s Rule to solve systems of linear functions in two and three variables. l Use determinants to calculate the area of triangles in the coordinate plane. l Use determinants to determine the equation of a line passing through two points. Key Terms l Cramer s Rule l determinant Problem Cramer s Rule 200 Carnegie Learning, Inc. Gabriel Cramer (04 52) was a Swiss mathematician. He discovered Cramer s Rule, a method of solving a system of linear equations. Cramer s method uses an operation that transforms a square matrix of numbers into a single value, called the determinant of the square matrix. a a 2 a 2 a 22 The determinant of a 2 2 matrix is calculated as: det a a 2 a 2 a 22 a a 2 a 2 a 22 a a 22 a 2 a 2 Or using simpler notation: a b c d ad bc. Calculate the determinant of each 2 2 array. a b c. 6 d Lesson.6 l Determinants 45
46 e f To use Cramer s Rule to solve a system of linear equations in two variables, write each equation in standard form. ax by c dx ey f Calculate three determinants. Notice that D is the determinant of the array consisting of the coefficients of the variables. D x is the same as D, except the x coefficients are replaced by the constants. D y is the same as D, except the y coefficients are replaced by the constants. D a d b e D x c f b e D y a d c f The solution to the system of equations is x D x D and y D y D 2. Solve each system of equations using Cramer s Rule. 2x y 6 a. x 2y 2 for D Carnegie Learning, Inc. 46 Chapter l Matrix Operations and Equations
47 x y 6 b. x y 0 2.2x 4.y 6. c..2x.2y Carnegie Learning, Inc. Lesson.6 l Determinants 4
48 d. 2 2 x 5 y 5 x y 2 Problem 2 Determinants of by Matrices One method for calculating the determinant of a matrix involves rewriting the first two columns, and then calculating the determinant as shown. a a 2 a a a 2 a 2 a 22 a 2 a 2 a 22 a a 2 a a a 2 a a 22 a a a 2 a 2 a a 2 a a a 22 a a 2 a a a a 2 a 2 (a a 22 a a 2 a 2 a a a 2 a 2 ) (a a 22 a a a 2 a 2 a 2 a 2 a ) For example, calculate the determinant of the array as shown ( 20 6) (4 0 24) A second method for calculating the determinant of a matrix involves minor determinants as shown. a a 2 a a 2 a 22 a a 2 a a 2 a a a 22 2 a 2 a a 2 a a 2 2 a a a a 2 a 22 a a 2 a (a 22 a a 2 a 2 ) a 2 (a 2 a a 2 a ) a (a 2 a 2 a 22 a ) 200 Carnegie Learning, Inc. a a 22 a a a 2 a 2 a 2 a 2 a a 2 a 2 a a a 2 a 2 a a 22 a (a a 22 a a 2 a 2 a a a 2 a 2 ) (a a 22 a a a 2 a 2 a 2 a 2 a ) 48 Chapter l Matrix Operations and Equations
49 For example, calculate the determinant of the array as shown (2) ( 0) 2(2 (0)) 2(8 2) Calculate each determinant by rewriting the first two columns. a b Calculate the value of each determinant using the minor determinants. 2 a Carnegie Learning, Inc. b Lesson.6 l Determinants 49
50 . Solve the system of equations using Cramer s Rule. 2x y z 6x 2y z 2 4x y 2z 200 Carnegie Learning, Inc. 50 Chapter l Matrix Operations and Equations
51 Problem Area of Triangles, Collinear Points, and Equations of Lines Determinants can be used to calculate the area of a triangle in the coordinate plane using the coordinates of the vertices. The area of the triangle with vertices (a, b), (c, d), and (e, f ) is 2 a c e b d f (a(d f ) b(c e) (cf de)) 2 The area must be positive. If the determinant is negative, then multiply by 2. If the determinant is positive, then multiply by 2.. Determine the area of each triangle. a. A triangle has vertices (2, ), (5, 6), and (, 8). b. y Carnegie Learning, Inc x Lesson.6 l Determinants 5
52 2. Consider a triangle with vertices at (2, 9), (2, ), and (0, 5). a. Calculate the area of the triangle. b. What does the value of the determinant tell you about the three points? Explain. c. What rule is suggested for the points (a, b), (c, d ), and (e, f )?. Consider the points (x, y), (2, ) and (0, 5). a. What can you conclude about the points if x 2 0 y 5 0? b. Rewrite the determinant equation by evaluating the determinant. 200 Carnegie Learning, Inc. c. What does the rewritten equation mean in terms of the original points? 52 Chapter l Matrix Operations and Equations
53 d. Verify your conclusion in part (c). e. What rule is suggested for the points (x, y), (c, d ) and (e, f )? 4. Determine the equation of the line that passes through the points: a. (, 2) and (, 4) b. (, 2) and (9, 2) 200 Carnegie Learning, Inc. Be prepared to share your solutions and methods with the entire class. Lesson.6 l Determinants 5
54 200 Carnegie Learning, Inc. 54 Chapter l Matrix Operations and Equations
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