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3-5 Perform Basic Matrix Operations Name:_____________________ Objective: To perform basic operations with matrices. Algebra 2 Standard 2.0 *A matrix is a rectangular arrangement of numbers in __________ and ___________________. 4 1 5 Ex. 0 6 3 *Two matrices are equal if their dimensions are the same and the elements in the corresponding positions are equal. *To add or subtract two matrices, simply add or subtract corresponding elements. You can add or subtract matrices only if they have the _____________ dimensions. a b e c d g f ae b f h c g d h a b e Subtracting Matrices c d g f a e b f h c g d h Adding Matrices * Perform the indicated operation, if possible. Ex. 1: 2 4 3 2 a. 0 1 4 0 You Try: 2 11 3 1 a. 4 5 2 8 4 6 2 3 b. 2 2 4 2 1 5 3 1 2 4 0 2 b. 7 2 3 0 3 1 5 14 *Scalar Multiplication: Ex. 2: Perform the indicated operation, if possible. a. 3 2 4 0 3 1 6 1 5 2 1 b. 3 4 0 5 3 You Try: 2 1 3 a. 4 7 6 1 2 0 5 Algebra 2 Ch.3B Notes Page 1 4 1 2 2 b. 3 3 5 0 6 *Properties of Matrix Operations: Let A, B, and C be matrices with the same dimensions, and let k be a scaler. _____________________Property of Addition: A B C A B C _____________________Property of Addition: A B B A _____________________Property of Addition: k A B kA kB _____________________Property of Subtraction: k A B kA kB Ex. 3; A local bakery keeps track of their sales as shown below. Last Month: Store 1: 650 rolls, 220 cakes, 32 pies; Store 2: 540 rolls, 200 cakes, 30 pies This Month: Store 1: 840 rolls, 250 cakes, 50 pies; Store 2: 800 rolls, 250 cakes, 42 pies Organize the data using matrices. Then write and interpret a matrix giving total number of bakery items sold per store. Ex. 4: Solve the matrix equation for x ad y. 2x 1 2 3 4 4 4 y 1 4 3 6 Ex. 5: Solve the matrix equation for x ad y. 2 x 3 1 4 10 2 2 5 y 3 5 16 14 You Try: Solve the matrix equation for x ad y. 3x 1 9 4 12 10 2 y 5 3 2 18 4 Algebra 2 Ch.3B Notes Page 2 3-6 Multiply Matrices Name:____________________ Objective: To multiply matrices. Algebra 2 Standard 2.0 * The product of two matrices A and B is defined (possible) when the number of __________ in A is ________________ to the number of _____________ in B. A B AB ( m n) ( n p ) ( m p ) Ex. 1: State whether the product is defined. If so, give the dimensions of AB. a. A: 3×5, B: 5×2 b. A: 3×4, B: 3×2 You Try: State whether the product is defined. If so, give the dimensions of AB. a. A: 5×2, B: 2×2 b. A: 3×2, B: 3×2 A *Multiplying Matrices: a b e c d g B f ae bg h ce dg 2 3 Ex. 2: Find AB if A = and B = 1 5 af bh cf dh 1 4 . 3 2 3 3 You Try: Find AB if A = and B = 1 2 Algebra 2 Ch.3B Notes Page 3 AB 1 5 . 3 2 * Using the given matrices, evaluate the expression. 3 2 2 3 2 1 A 0 4 ; B ; C 1 0 4 2 1 5 Ex. 3: A(B-C) You Try: A(B+C) *Properties of Matrix Multiplication: Let A, B, and C be matrices and k be a scalar. Associative Property of Matrix Multiplication: ________________________________ Left Distributive Property :________________________________________________ Right Distributive Property :_______________________________________________ Associative Property of Scalar Multiplication: _________________________________ Inventory Cost per item Total Cost matrix Cost Matrices: matrix matrix m n n p m p Ex. 4: The matrix below represents the inventory for a chain of entertainment stores. The first row is Store 1, the second row is Store 2, and the third row is Store 3. CDs DVD VHS Games 2800 550 200 150 2600 800 150 120 1850 650 190 100 If CD’s cost $15, DVDs cost $20, VHSs cost $18, and Games cost $30, find the total value of the inventory for each store. Algebra 2 Ch.3B Notes Page 4 3-7 Evaluate Determinants Name:_________________ Objective: To evaluate determinants of matrices. Algebra 2 Standard 2.0 *Determinant: A real number associated with any square (n×n) matrix. The determinant of a matrix A is denoted by _____________ or by _____________. a b a b *Determinant of a 2×2 Matrix: det ad cb c d c d *Determinant of a 3×3 Matrix: a b c a b c det d e f d e f (aei bfg cdh) ( gec hfa idb) g h i g h i Ex 1: Evaluate the determinants: 6 2 a. 1 4 4 2 0 b. 1 1 2 2 5 3 10 2 3 c. 2 12 4 0 7 2 You Try: Evaluate the determinants: 4 1 2 3 2 a. b. 3 2 1 6 1 0 5 1 *Area of a Triangle: The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by x1 1 Area x2 2 x3 y1 1 y2 1 y3 1 where the symbol ± indicates that the appropriate sign should be chosen to yield a positive value. Algebra 2 Ch.3B Notes Page 5 Ex. 2: The approximate coordinates (in miles) of a triangular region representing a city and its suburbs are (10, 20), (-8, 5) and (-4, -5). Find the area of the region. 10 20 1 1 Area 8 5 1 2 4 5 1 You Try: Find the area of the triangle shown. Algebra 2 Ch.3B Notes Page 6 3-8A Use Inverse Matrices to Solve Linear Systems Name:____________ Objective: To find the inverse of 2×2 matrices. Algebra 2 Standard 2.0 * The n×n identity matrix is a matrix with ____ on the main diagonal and ____ elsewhere. If A is any n×n matrix and I is the n×n ___________ matrix, then AI =A and IA=A. 2×2 Identity Matrix 1 0 I 0 1 3×3 Identity Matrix 1 0 0 I 0 1 0 0 0 1 * Two n×n matrices A and B are inverse of each other if their __________ (in both orders) is the n×n ________________ matrix. That is, AB = I and BA = I. An n×n matrix has an inverse if and only if det A ≠ 0. The symbol for the inverse of A is A-1. The inverse of a 2×2 matrix: a b The inverse of the matrix A is c d A1 1 A d b 1 d b c a ad bc c a provided ad bc 0 5 2 Ex. 1a: Find the inverse of A 8 3 6 1 Ex. 1b: Find the inverse of A 2 4 1 5 You Try: Find the inverse of A 4 8 Algebra 2 Ch.3B Notes Page 7 *Solve a matrix equation: 1. Find the inverse matrix of A. 2. Multiply each side of AX = B by A-1 on the ____________ to find the solution X = A-1B. Ex. 2a: Solve the matrix equation AX = B for the 2×2 matrix X. 3 2 2 4 X 7 5 3 1 Ex. 2b: Solve the matrix equation AX = B for the 2×3 matrix X. 5 2 4 5 0 X 9 3 3 1 6 You Try: Solve the matrix equation AX = B for the 2×2 matrix X. 4 1 8 9 X 0 6 24 6 Algebra 2 Ch.3B Notes Page 8 3-8B Use Inverse Matrices to Solve Linear Systems Name________________ Objective: To solve linear systems using inverses. Algebra 2 Standard 2.0 * The inverse of a 3×3 matrix is very difficult to compute by hand, so we will use a calculator. * Use a graphing calculator to find the inverse of A. Then use the calculator to verify the result by showing that AA-1 = I and A-1A =I Ex. 1: Ex. 2: 1 A 3 1 2 A 2 12 You Try: 0 2 2 1 1 4 2 2 6 0 0 4 2 A 5 4 1 2 3 0 3 8 *Using Inverse Matrices to Solve Linear Systems: 3. Write the system as a matrix equation Ax = B. The matrix A is the coefficient matrix, X is the matrix of variables, and B is the matrix of constants. 4. Find the inverse matrix of A. 5. Multiply each side of AX = B by A-1 on the ___________ to find the solution X = A-1B. Ex. 3: Use an inverse matrix to solve the linear system. 5x – 2y = -10 2x – 4y = 12 Algebra 2 Ch.3B Notes Page 9 Ex. 4: Use an inverse matrix to solve the linear system. 2 x y 6 z 4 6 x 4 y 5 z 7 4 x 2 y 5 z 9 You Try: Use an inverse matrix to solve the linear system. 3 x y 2 z 10 6 x 2 y z 2 x 4 y 3z 7 Ex. 5: At a video store, the cost of 3 DVDs, 2 video games, and 4 VHS movies is $130. The cost of 2 DVDs, 1 video game, and 5 VHS movies is $105. The cost of 3 DVDs, 3 video games, and 3 VHS movies is $135. Find the cost of each item. Algebra 2 Ch.3B Notes Page 10