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Chapter 4 Matrices In Chapter 4, You Will… • Move from using matrices in organizing data to manipulating matrices through data. • Learn to represent real-world relationships by writing matrices and using operations such as addition and multiplication to develop new matrices. 4-1 Organizing Data What you’ll learn … • To identify matrices and their elements • To organize data into matrices 1.04 Operate with matrices to model and solve problems. A matrix (plural matrices) is a rectangular array of numbers written within brackets. The number of horizontal rows and the number of vertical columns determine the dimensions of a matrix. Columns Rows Example 1 Writing the Dimensions of a Matrix Write the dimensions of each matrix. Each number in a matrix is a matrix element. You can identify a matrix element by its position within the matrix. Use a lowercase letter with subscripts. The subscripts represent the element’s row number and column number. Consider the matrix The element a21 = 1, since the element in the 2nd row and 1st column is 1. The element a13 = 9, since the element in the 1st row and 3rd column is 9. Example 2 Identifying a Matrix Element Identify each matrix element 17 10.4 9 a. a33 24 12 30 b. a11 3 15 15 c. a21 d. a12 Example 4a Real World Example Gymnast Floor Exercise Vault Balance Beam Uneven Bars Amy Chow 9.525 9.468 9.625 9.400 Dominique Dawes 9.087 9.393 8.600 9.675 Kristin Maloney 9.525 9.225 9.312 9.575 Elise Ray 9.225 9.468 9.687 9.687 • Which element represents Maloney’s score on the vault? • Write a matrix W to represent the information. Example 4b Real World Example 4000 3900 3400 3500 3000 2500 Greece Canada USA 2000 1500 1000 500 0 400 1985-1989 50 100 1990-1994 25 Example 4b Real World Example Continued • Write a matrix M to represent the data in the graph, with columns representing years. • What are the dimensions of this matrix? • What does the first row represent? • What does m32 represent? 4-2 Adding and Subtracting Matrices What you’ll learn … • To add and subtract matrices. • To solve certain matrix equations. 1.04 Operate with matrices to model and solve problems. Adding and Subtracting Matrices You must perform matrix addition or subtraction on matrices with equal dimensions by adding or subtracting the corresponding elements, which are elements in the same position in each matrix. Example 1 -2 0 3 -5 7 -12 24 -3 5 -1 10 Adding Matrices 3 9 -3 -9 6 12 -3 1 2 -4 -1 5 The additive identity matrix for the set of all m x n matrices is the zero matrix 0, or Omxn ,whose elements are all zeros. The opposite, or additive inverse, of an m x n matrix A is –A. -A is the m x n matrix with elements that are the opposites of the corresponding elements of A. A= 2 -4 -1 0 A + (-A) = 0 A+0=A -A = -2 4 1 0 Example 2 Using Identity and Inverse Matrices 1 2 5 -7 0 0 0 0 2 8 -3 0 -2 -8 3 0 Properties of Matrix Addition If A, B, and C are m x n matrices, then • A + B is an m x n matrix Closure Property • A+B=B+A Commutative Property of Addition • (A+B)+C = A+(B+C) Associative Property of Addition • There exist a unique m x n matrix O such that O+A=A+O=A. Additive Identity Property • For each A, there exists a unique opposite, -A. A+(-A)=0 Additive Inverse Property Subtracting Matrices 1 -2 0 3 -5 7 3 9 -3 -9 6 12 Just Add the Opposite 1 -2 0 3 -5 7 -3 -9 3 9 -6 -12 Example 3 Subtracting Matrices 6 -9 7 -2 1 8 -4 3 0 6 5 10 -3 5 -1 -10 -3 1 2 -4 A matrix equation is an equation in which the variable is a matrix. You can use the addition and subtraction properties of equality to solve matrix equations. X + -1 0 2 5 10 7 = -4 4 Equal matrices are matrices with the same dimensions and equal corresponding elements. Example 4 Solving A Matrix Equation • Solve X - 1 1 = 0 1 • Solve X + -1 0 = 10 7 2 5 -4 4 3 2 8 9 Example 5 Determining Equal Matrices Determine whether the two matrices in each pair are equal. -0.75 1/5 ½ -2 4 6 8 -3/4 0.2 0.5 -2 8/2 18/3 16/2 Example 6 Finding Unknown Matrix Elements 2x-5 4 3 3y+12 x+8 -5 3 -y 25 4 3 y+18 38 -5 3 4y-10 X = ____ Y = ____ X = ____ Y = ____ 4-3 Matrix Multiplication What you’ll learn … • To multiply a matrix by a scalar • To multiply two matrices 1.04 Operate with matrices to model and solve problems. You can multiply a matrix by a real number. 3 5 = 3 2 8 9 15 6 24 The real number factor (such as 3) is called a scalar. You find the scalar product by multiplying each element of the matrix by the scalar. Example Scalar Multiplication 15 -12 10 0 -3 20 -10 7 0 Example 2 Using Scalar Products 2 3 -7 A= 1 4 5 Find 5B- 4A 3 0 6 B= -1 8 2 Find A + 6B Properties of Scalar Multiplication If A, B, and O are m x n matrices and c and d are scalars, then • cA is an m x n matrix Closure property • (cd)A = c(dA) Associative Property of Multiplication • C(A+B) = cA+cB (c+d)A = cA + cb • 1 (A) = A • 0(A) = c0 = 0 Distributive Property Multiplication Identity Property Multiplication Property of 0 Example 3a Solving Matrix Equations with Scalars 3 4 4x + 2 -2 1 = 10 0 4 2 Example 3b Solving Matrix Equations with Scalars -3x + 7 0 -1 2 -3 4 = 10 0 8 -19 -18 10 Investigation: Using Matrices 1. 2. 3. How much money did the cafeteria collect selling lunch 1? Selling Lunch 2? Selling Lunch 3? a. How much did the cafeteria collect selling all 3 lunches? b. Explain how you used the data in the table to find your answer. a. Write a 1x3 matrix to represent the cost of the lunches. b. Write a 1x 3 matrix to represent the number of lunches sold. c. Describe a procedure for using your matrices to find how much money the cafeteria collected from selling all three lunches. Lunch 1 Lunch 2 Lunch 3 Cost per Lunch $2.50 $1.75 $2.00 Number Sold 50 100 75 To perform matrix multiplication, multiply the elements of each row of the first matrix by the elements of each column of the second matrix. Add the products. **Multiply rows times columns. **You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix. 8 2 3 2 5 1 5 7 1 0 0 3 They must match. Dimensions: 2x3 3x2 The dimensions of your answer. **Multiply rows times columns. **You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix 3 9 2 2 1 2. 5 7 6 3 4 Dimensions: 2 x 3 2x2 *They don’t match so can’t be multiplied together.* 0 -1 1 0 *Answer should be a 2 x 2 4 -3 -2 5 2x2 2x2 0(4) + (-1)(-2) 0(-3) + (-1)(5) 1(4) + 0(-2) 1(-3) +0(5) 2 -5 4 -3 Example: -2 5 3 -1 -2(4)+5(2) 4 -4 2 6 -2(-4)+5(6) 3(4) + -1(2) 3(-4) + -1(6) 2 38 10 -18 Example 4a Multiplying Matrices • Find the product of -1 0 3 -4 -3 3 5 0 • Multiply a11 and b11. Add the products. • The result is the element in the first row and first column. • Repeat with the rest of the rows and columns. Example 4b Multiplying Matrices • Find the product of -3 3 5 0 -1 0 3 -4 • Multiply a11 and b11. Add the products. • The result is the element in the first row and first column. • Repeat with the rest of the rows and columns. Example 5 Real World Connection A used-record store sells tapes, LP records, and compact discs. The matrices show today’s information. Find the store’s gross income for the day. Tapes $8 LPs $6 CDs $13 Tapes LPs CDs 9 30 20 Example 5 Find each product. 12 3 10 -5 10 -5 12 3 0 0 Example 6a Determining When a Product Matrix Exists 8 0 2 3 Use matrices G = -1 8 and H = 2 -5 . 4 0 Determine whether products GH and HG are defined (exist) or undefined (do not exist). Example 6b Determining When a Product Matrix Exists Use matrices R = 4 -2 and S = 5 -4 8 0 -1 0 2 -5 . 1 8 Determine whether products RS and SR are defined (exist) or undefined (do not exist). Properties of Matrix Multiplication If A, B, and C are n x n matrices, then • AB is an n x n Closure Property • (AB)C = A(BC) • • A(B+C) = AB + BC (B+C)A = BA + CA Associative Property of Multiplication Distributive Property • OA = AO = O, where O has the same dimensions as A. Multiplicative Property of 0 4-5 2x2 Matrices, Determinants, and Inverses What you’ll learn … • To evaluate determinants of 2x2 matrices and find inverse matrices • To use inverse matrices in solving matrix equations 1.04 Operate with matrices to model and solve problems. Evaluating Determinants of 2x2 Matrices • A square matrix is a matrix with the same number of columns as rows. • For an n x n square matrix, the multiplicative identity matrix is an n x n square matrix I, with 1’s along the diagonal and zeros elsewhere. Multiplicative Inverse of a Matrix If A and X are n x n matrices, and AX=XA=I, then X is the multiplicative inverse of A, written A-1. A -1 (A ) = -1 A (A) =I Example 1 Verifying Inverses A= 2 3 1 2 3 -1 M= 7 -1 B= 2 -3 -1 2 N = .1 .1 -.7 .3 Determinant of a 2x2 Matrix The determinant of a 2x2 matrix is ad – bc. a b c d -5(-3) – 7(2) 15 – 14 1 Example 2 Evaluating the Determinants of a 2x2 Matrix 4 2 det 4 2 det 8 7 det 2 3 k 3 3-k -3 Inverse of a 2x2 Matrix Let A = a b c d . If det A ≠0 then A has an inverse. If det A ≠0, then A-1 = 1__ __1__ = = detA ad - bc d -b -c a Example 3 Finding an Inverse Matrix 2 4 1 3 12 9 4 3 .5 2.3 3 7.2 6 5 25 20 Example 4 Solving a Matrix Equation -2 -5 X = -2 2 1 3 3 4 -4 -5 0 -22 X = 0 -28 AX = B A-1AX = A-1B IX = A-1B X= A-1B 4-6 3x3 Matrices, Determinants, and Inverses What you’ll learn … • To evaluate determinants of 3x3 matrices 1.04 Operate with matrices to model and solve problems. Matrix Determinants A Determinant is a real number associated with a matrix. Only SQUARE matrices have a determinant. The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A| or 3 1 . 2 4 Matrix Determinants To find the determinant of a 3 x 3 matrix, first recopy the first two columns. Then do 6 diagonal products. 18 60 16 5 2 6 5 2 2 1 4 2 1 3 3 4 3 3 -20 -24 36 94 - (-8) 94 +8 = 102 Step 1: Rewrite first two columns of the matrix. Step 2: multiply diagonals going up! Step 3: multiply diagonals going down! Step 4: Bottom minus top! Example 1 Evaluating the Determinant of a 3x3 Matrix -1 2 0 3 -4 1 5 6 -1 Step 1: Rewrite first two columns of the matrix. Step 2: multiply diagonals going up! Step 3: multiply diagonals going down! Step 4: Bottom minus top! Now You Are Asking … How can I do this on the calculator? • Step 1: Go to Matrix (above the x-1 key) • Step 2: Arrow to the right to EDIT to allow for entering the matrix. • Step 3: Type in the dimensions (size) of your matrix and enter the values (press ENTER). • Step 4: Go to Matrix again. Notice that your matrix (3x3) now appears • showing it in residence. • Step 5: Arrow to the right to MATH. Choose #1: det ( • Step 6: The function det( will appear on the home screen waiting for a parameter (in this case, the name of the determinant to evaluate). • Step 7: Go to Matrix again. Choose [A] or whichever location holds your matrix. • Step 8: The name of the determinant's location will appear as the parameter. Hit ENTER to see the evaluation. Example 2 Using a Calculator -1 3 2 -4 0 1 5 6 -1 1 0 3 2 2 10 -1 4 -6 Step 1: Enter matrix A into your calculator. Step 2: Use the matrix submenus to evaluate the determinant of A. Identity Matrices An identity matrix is a square matrix that has 1’s along the main diagonal and 0’s everywhere else. 1 0 0 1 0 0 1 0 0 1 0 0 1 When you multiply a matrix by the identity matrix, you get the original matrix. Inverse Matrices When you multiply a matrix and its inverse, you get the identity matrix. If A and B are inverse matrices, then AB=BA=I. Matrix A Inverse of Matrix A Identity 3 1 2 1 1 0 X 5 2 5 3 0 1 Example 4 Solving a Matrix Equation 0 1 1 0 3 -2 2 0 1 0 1 0 2 -2 1 1 4 0 X = X = 0 -6 19 -1 8 -2 Use the equation X = A-1C 4-7 Inverse Matrices and Systems What you’ll learn … • To solve systems of equations using inverse matrices. 1.04 Operate with matrices to model and solve problems. 2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties. You can represent a system of equations with a matrix equation. System of Equation x + 2y = 5 3x +5y = 14 Matrix Equation 1 3 2 5 x 5 y = 14 Example 1 and 2 Writing and Solving a System as a Matrix. System of Equation 5a + 3b = 7 3a + 2b = 5 System of Equation x + 3y = 22 3x + 2y = 10 = = Example 3 Solving a System of Three Equations. System of Equation 2x + y +3z = 1 5x + y – 2z = 8 x – y -9z = 5 System of Equation x–y+z=0 x – 2y – z = 5 2x - y + 2z = 8 = = Example 5 Unique Solutions. System of Equation x+y=3 x–y=7 System of Equation 3x + 5y = 1 2x –y = -8 Remember….. a b = ad - bc det c d In Chapter 4, You Should Have… • Moved from using matrices in organizing data to manipulating matrices through data. • Learned to represent real-world relationships by writing matrices and using operations such as addition and multiplication to develop new matrices.