Think about systems of linear equations, their solutions, and how they might be represented with matrices.

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1 Think About This Situation Unit 4 Lesson 3 Investigation 1 Name: Think about systems of linear equations, their solutions, and how they might be represented with matrices. a Consider a system of two linear equations like the one below. x - 2y = 8 3x + 6y = 18 i. What, if anything, can you say about the solution of this system just by examining the equations? ii. What are some methods you know for solving such a system? iii. Which method would you use? Why? b Can you think of a way you might represent the system of equations above with matrices and matrix multiplication? The first step in learning how to use matrices to solve linear systems is to examine more closely some properties of matrices. In Investigation 1, you will learn about some important properties of matrices and compare those properties to properties of real numbers. In Investigation 2, you will learn how to represent a system of two linear equations as a matrix equation and then how to solve the matrix equation. Finally, in Investigation 3, you will compare this matrix method for solving linear systems to methods you have previously learned. Investigation 1 Properties of Matrices Matrices and matrix operations obey certain properties. The matrix methods you use to solve problems often depend upon these properties. As you investigate properties of matrices, look for answers to these questions: What are some important properties of operations with matrices? How are these properties similar to, and different from, properties of operations with real numbers? In arithmetic, you studied numbers and operations on numbers. In algebra, you have studied expressions and operations on expressions. In both settings, you found that certain properties were obeyed. For example, one such property is the Associative Property of Addition: a + (b + c) = (a + b) + c. In this unit, you have been studying matrices and matrix operations, including matrix addition and matrix multiplication. You will now investigate properties of matrices and their operations. This situation occurs frequently in mathematics certain mathematical objects (like numbers or matrices) along with operations on those objects are studied and then their properties are examined. LESSON 3 Matrices and Systems of Linear Equations 133

2 1 Matrix Addition To begin this exploration of properties of matrices, first consider matrix addition. a. Suppose A = To which of the following matrices can A be added? B = C = D = E = b. Under what conditions can two matrices be added? State the conditions as precisely as you can and explain your reasoning. 2 Commutative Property of Addition You know from your previous studies that the order in which you add two numbers does not matter. That is, for all real numbers a and b, a + b = b + a. a. Give three examples of the Commutative Property of Addition for real numbers. b. Suppose A = and B = Is it true that A + B = B + A? c. Do you think A + B = B + A for all matrices A and B (assuming that A has the same number of rows and columns as B)? Defend your answer. 3 Additive Identity The number 0 has a unique property with respect to addition: adding 0 to any real number leaves the number unchanged. That is, a + 0 = a, for all real numbers. The number 0 is called the additive identity. Consider a similar situation for matrices. 4 2 a. Suppose A = Find a matrix C so that A + C = A. 8 7 b. Suppose matrix B has 4 rows and 3 columns. Find a matrix E such that B + E = B. c. Look at the matrices you found in Parts a and b. Write a description (definition) of the additive identity matrix for m n matrices. Such a matrix is also called a zero matrix. 4 Additive Inverse Every real number has an additive inverse. A number and its additive inverse sum to zero. a. What is the additive inverse of 17? Of _ 3? Of ? 4 b. A matrix and its additive inverse matrix sum to the zero matrix. Let C = Find the additive inverse matrix for C by filling in the blanks for the matrix below: C + = c. For any matrix A, describe (define) the additive inverse matrix for A. 2_ UNIT 2 Matrix Methods

3 5 Matrix Multiplication Another important matrix operation is matrix multiplication. As you discovered in Lesson 2, only matrices of compatible sizes can be multiplied. a. Suppose A is a matrix that has 4 rows and 2 columns, and matrix B has 3 rows and 4 columns. Is it possible to multiply A B? How about B A? Explain. b. Suppose C is a matrix that can be multiplied on the right by a 3 2 matrix, D. That is, you can multiply C D. What could be the size of C? What would be the size of the product matrix? c. Suppose matrix A has size m n. What must be the size of B so that it is possible to multiply A B? What must be the size of the product matrix? d. Sometimes it is possible to multiply two matrices in either order. What are the conditions on the sizes of two matrices A and B so that it is possible to multiply A B and also B A? 6 Commutative Property of Multiplication In the case of real numbers, you know that the order of multiplication of numbers does not matter. That is, ab = ba, for all real numbers a and b. a. Give three examples illustrating the Commutative Property of Multiplication for real numbers. b. Check to see if the commutative property is true for multiplication of 2 2 matrices. Explain your reasoning. Compare with other students and resolve any differences. c. In Part d of Problem 5, you found a condition on the sizes of two matrices A and B so that it is possible to multiply both A B and also B A. But just because it is possible to multiply in both orders, do you necessarily get the same answer? You just explored this question with 2 2 matrices in Part b. Check it out for some other size matrices of your choice. For example, construct a 2 3 matrix and a 3 2 matrix, then multiply in both orders and see if you get the same answer. d. Based on your work above, is the commutative property true for matrix multiplication? Explain. Recall that a matrix with the same number of rows and columns is called a square matrix. Square matrices have several important properties with respect to matrix multiplication. 7 Multiplicative Identity The number 1 has the unique property that multiplying any real number by 1 does not change the number. That is, a 1 = 1 a = a, for all real numbers a. The number 1 is called the multiplicative identity. A square matrix that acts like the number 1 in this regard is called an identity matrix (or multiplicative identity matrix). Multiplying a matrix by the identity matrix does not change the matrix. That is, an identity matrix I has the property that A I = I A = A. Identity matrices are always square. LESSON 3 Matrices and Systems of Linear Equations 135

4 a. Find the identity matrix for 2 2 square matrices by filling in the blanks for the matrix below. 5 4 = 5 4 Compare your answer with those of other students. Resolve any differences. b. Multiply 5 4 on the left by the identity matrix you found in Part a. Check that you get 5 4 as the answer. c. Suppose matrix A has 3 rows and 3 columns. Find the identity matrix I such that A I = A. d. Write a description of an identity matrix. 8 Multiplicative Inverse The product of a number and its multiplicative inverse is 1. Every nonzero number has a multiplicative inverse. For example, the multiplicative inverse of 5 is _ 1 5 since 5 _ 1 5 = 1. a. What is the multiplicative inverse of 3? Of _ 1 2? Of _ 5 3? b. Just as the product of a number and its multiplicative inverse is 1, the product of a matrix and its multiplicative inverse matrix is I. That is, the multiplicative inverse matrix for the square matrix D is the matrix written D -1, such that D D -1 = I, where I is the identity matrix. Suppose D = 5 3. Make and test a conjecture about the entries of D -1. c. There are several systematic methods for finding the entries of D -1. One way is to use the fact that if D -1 is a matrix a b c d such that D D -1 = I, then 5 3 a b c d = Test the following strategy for finding numbers a, b, c, and d that make this matrix equation true. Step 1: Perform the indicated matrix multiplication to create a system of four linear equations. 5 3 a b c d = One equation is 5a + 3c = 1. Write down the other three equations. 136 UNIT 2 Matrix Methods

5 Step 2: Solve the system of two equations involving a and c for a and c. Then solve the other system of two equations for b and d. Step 3: Use the results from Step 2 to write D -1. Check that D D -1 = I. 9 A multiplicative inverse matrix is often simply an inverse matrix. Other methods for finding an inverse matrix include using technology, using a formula, and using special matrix manipulations. A particular method using technology is provided below. (Other methods are examined in the On Your Own tasks.) a. Consider again the matrix D = 5 3. Compute the inverse matrix for D using your calculator or computer software. On most calculators, this can be done by entering matrix D into your calculator and then pressing the x 1 key. Compare this matrix with what you found in Problem 8. Resolve any differences. b. An inverse matrix should work whether multiplied from the right or the left. That is, D D -1 = D -1 D = I. i. For matrix D from Part a, check that D -1 D = I. ii. Also check that D D -1 = I. c. Use your calculator or computer software to find A -1, where A = Check that A -1 A = A A -1 = I. CPMP-Tools 10 Every real number (except 0) has a multiplicative inverse. Check to see if square matrices have this property. a. Consider A = 0 9. Without using your calculator or computer 0 4 software, try to find entries a, b, c, and d that will make the matrix equation true. a b c d = Does A have an inverse? That is, does the matrix A -1 exist? b. Find a square matrix with all nonzero entries that does not have an inverse. LESSON 3 Matrices and Systems of Linear Equations 137

6 Summarize the Mathematics In this investigation, you examined properties of matrices and their operations and compared them with corresponding properties of real numbers. a What are the conditions on the sizes of two matrices that allow them to be added? Describe the size of the sum matrix. b What are the conditions on the sizes of two matrices that allow them to be multiplied? Describe the size of the product matrix. c Describe and give an example of each of the following: i. a matrix and its additive inverse ii. a (multiplicative) identity matrix iii. a square matrix and its (multiplicative) inverse iv. a square matrix that does not have a (multiplicative) inverse v. two matrices A and B for which A B and B A are both defined, but A B B A d List some properties of real numbers and their operations that are shared by matrices and their operations. e List some properties of real numbers that are not shared by matrices. Be prepared to share your descriptions, examples, and thinking with the class. Check Your Understanding Investigate other similarities and differences between operations on real numbers and the corresponding operations on matrices. a. An important property of multiplication of numbers concerns products that equal zero. If x and y are real numbers and if xy = 0, what can you conclude about x or y? Is it possible that xy = 0, and yet x 0 and y 0? b. Do you think the property in Part a is true for matrix multiplication? Make a conjecture, and then consider Part c below. c. Suppose A = and B = Compute A B. Is it true for matrices that if A B = 0, then either A = 0 or B = 0? d. Think of another property of addition or multiplication of real numbers, and investigate whether matrices also have this property. Prepare a brief summary of your findings. 138 UNIT 2 Matrix Methods