Perform Basic Matrix Operations

Similar documents
Add, Subtract, and Multiply Polynomials

Evaluate and Simplify Algebraic Expressions

Solve Linear Systems Algebraically

A linear inequality in one variable can be written in one of the following forms, where a and b are real numbers and a Þ 0:

Solve Systems of Linear Equations in Three Variables

Solve Quadratic Equations by Completing the Square

Model Direct Variation. You wrote and graphed linear equations. You will write and graph direct variation equations.

Graph Quadratic Functions in Standard Form

Algebra II Notes Unit Four: Matrices and Determinants

Solve Absolute Value Equations and Inequalities

Analyze Geometric Sequences and Series

Evaluate and Graph Polynomial Functions

Graph Square Root and Cube Root Functions

Find Sums of Infinite Geometric Series

Solve Exponential and Logarithmic Equations. You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations.

Apply Properties of Logarithms. Let b, m, and n be positive numbers such that b Þ 1. m 1 log b. mn 5 log b. m }n 5 log b. log b.

Words Algebra Graph. 5 rise } run. } x2 2 x 1. m 5 y 2 2 y 1. slope. Find slope in real life

Graph Linear Inequalities in Two Variables. You solved linear inequalities in one variable. You will graph linear inequalities in two variables.

Apply Properties of 1.1 Real Numbers

You evaluated powers. You will simplify expressions involving powers. Consider what happens when you multiply two powers that have the same base:

Skills Practice Skills Practice for Lesson 1.1

Solve Radical Equations

Represent Relations and Functions

Write and Apply Exponential and Power Functions

Solving and Graphing Linear Inequalities 66.1 Solve Inequalities Using Addition and Subtraction

Five people were asked approximately how many hours of TV they watched per week. Their responses were as follows.

Solve Trigonometric Equations. Solve a trigonometric equation

Why? Sport. Source: The National Federation of State High School Associations. The element -1 is in Row 2, Column 1. 4 columns.

68% 95% 99.7% x x 1 σ. x 1 2σ. x 1 3σ. Find a normal probability

Algebra 1 End-of-Course Assessment Practice Test with Solutions

Goal: Write variable expressions and equations. a. A number increased by 3. c. 1 more than three times a number

Graph Simple Rational Functions. is a rational function. The graph of this function when a 5 1 is shown below.

Chapter 4: Systems of Equations and Inequalities

Algebra II Notes Matrices and Determinants Unit 04

Review for the Algebra EOC

Write Quadratic Functions and Models

Graph and Write Equations of Circles

Algebra 2 Level 2 Summer Packet

Practice A. Name Date. Evaluate the expression for the given value of the variable. Match the equation with its solution. Solve the equation.

Mathematics 1104B. Systems of Equations and Inequalities, and Matrices. Study Guide. Text: Mathematics 11. Alexander and Kelly; Addison-Wesley, 1998.

Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.

Solve Radical Equations

Algebra I Practice Exam

Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities

2.5 Justify a Number Trick

Essential Question How can you represent algebraic expressions using a coefficient matrix? A = [ 4 0

Properties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a

Unit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles

proportion, p. 163 cross product, p. 168 scale drawing, p. 170

College Algebra. Word Problems

Solving Linear Equations 33.1 Solve One-Step Equations

Solving Absolute Value Equations and Inequalities. The distance between. 0 and itself is 0, so 0 0.

Intensive Math-Algebra I Mini-Lesson MA.912.A.2.3

The Remainder and Factor Theorems

Monomial. 5 1 x A sum is not a monomial. 2 A monomial cannot have a. x 21. degree. 2x 3 1 x 2 2 5x Rewrite a polynomial

Grade 8. Functions 8.F.1-3. Student Pages

GEOMETRY (and Honors)

ALGEBRA 1 SEMESTER 1 INSTRUCTIONAL MATERIALS Courses: Algebra 1 S1 (#2201) and Foundations in Algebra 1 S1 (#7769)

14-3. Measures of Variability. OBJECTIVES Find the interquartile range, the semiinterquartile

Graph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses.

Unit Essential Questions. Can equations that appear to be different be equivalent? How can you solve equations?

6 th Grade - TNREADY REVIEW Q3 Expressions, Equations, Functions, and Inequalities

For Your Notebook E XAMPLE 1. Factor when b and c are positive KEY CONCEPT. CHECK (x 1 9)(x 1 2) 5 x 2 1 2x 1 9x Factoring x 2 1 bx 1 c

September 23, Chp 3.notebook. 3Linear Systems. and Matrices. 3.1 Solve Linear Systems by Graphing

0815AI Common Core State Standards

1. The area of the surface of the Atlantic Ocean is approximately 31,830,000 square miles. How is this area written in scientific notation?

Unit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles

Item Specification Sheet Algebra I Semester Exam

(2 x 2-3x + 5) + ( x 2 + 6x - 4) = 3 x 2 + 3x + 1 (continued on the next page)

due Thursday, August 25, Friday, September 2, 2016 test Prerequisite Skills for Algebra II Advanced

Analyzing Functions Maximum & Minimum Points Lesson 75

Grade 8 Mathematics MCA Item Sampler Teacher Guide

You studied exponential growth and decay functions.

Keystone Exam Review: Module 2. Linear Functions and Data Organizations

1.4 CONCEPT QUESTIONS, page 49

Algebra I Exam Review

Dividing Polynomials

Solve Quadratic Equations by Graphing

Unit 7 Systems and Linear Programming

10.7. Interpret the Discriminant. For Your Notebook. x5 2b 6 Ï} b 2 2 4ac E XAMPLE 1. Use the discriminant KEY CONCEPT

Algebra I Final Study Guide

Chapter 6 Outline Systems of Linear Inequalities

Unit 4 Linear Functions

Unit 1 Study Guide [MGSE9-12.N.Q.1-3, MGSE9-12.A.CED.1]

Instructional Materials for WCSD Math Common Finals

Algebra 1 S1 (#2201) Foundations in Algebra 1 S1 (#7769)

Minnesota K-12 Academic Standards in Mathematics (2007)

4-2. Matrix Addition. Vocabulary. How Are Matrices Added? Lesson. Definition of Matrix Addition. Mental Math

y in both equations.

Systems of Equations and Inequalities 77.1 Solve Linear Systems by Graphing

Matrix Operations (Adding, Subtracting, and Multiplying) Ch. 3.5, 6 Finding the Determinant and Inverse of a Matrix Ch. 3.7, 8

Create your own system of equations: 1. Prove (2, 5) is a solution for the following system: 2. Is (-2, 0) a solution for the following system?

Linear Functions, Equations, and Inequalities

Evaluate Logarithms and Graph Logarithmic Functions

Why? Step 3 Substitute the value from Step 2 into either equation, and solve for the other variable. Write the solution as an ordered pair.

Lesson 8: Representing Proportional Relationships with Equations

ax + b < c ax + b c Graphing Inequalities:

1Factor binomials that. 2Use the difference. Then. Why? Now. New Vocabulary dif ference of two squares

Study Guide and Review - Chapter 2. Choose the correct term to complete each sentence.

6-3 Rate of Change and Slope

Transcription:

TEKS 3.5 a.1, a. Perform Basic Matrix Operations Before You performed operations with real numbers. Now You will perform operations with matrices. Why? So you can organize sports data, as in Ex. 34. Key Vocabulary matrix dimensions elements equal matrices scalar scalar multiplication A matrix is a rectangular arrangement of numbers in rows and columns. For example, matrix A below has two rows and three columns. The dimensions of a matrix with m rows and n columns are m 3 n (read m by n ). So, the dimensions of matrix A are 3 3. The numbers in a matrix are its elements. A 5F 4 1 5 0 6 3G rows The element in the first row and third column is 5. 3 columns Two matrices are equal if their dimensions are the same and the elements in corresponding positions are equal. KEY CONCEPT For Your Notebook Adding and Subtracting Matrices To add or subtract two matrices, simply add or subtract elements in corresponding positions. You can add or subtract matrices only if they have the same dimensions. Adding Matrices F a b c dg 1F e f g hg 5F a 1 e b1 f c 1 g d1 hg Subtracting Matrices F a b c dg F e f g hg 5F a e b f c g d hg E XAMPLE 1 Add and subtract matrices AVOID ERRORS Be sure to verify that the dimensions of two matrices are equal before adding or subtracting them. Perform the indicated operation, if possible. a. F 3 0 5 1G 1F 1 4 0G 5F 3 1 (1) 0 1 4 5 1 1 1 0G 5F 4 b. F 7 6G 4 0 F 1 5 3 10 3 1G 3 1G 5F 1G 5F 7 () 4 5 9 0 3 (10) 1 (3) 6 1 3 8 5G 3.5 Perform Basic Matrix Operations 187

SCALAR MULTIPLICATION In matrix algebra, a real number is often called a scalar. To multiply a matrix by a scalar, you multiply each element in the matrix by the scalar. This process is called scalar multiplication. E XAMPLE Multiply a matrix by a scalar COMPARE ORDER OF OPERATIONS The order of operations for matrix expressions is similar to that for real numbers. In particular, you perform scalar multiplication before matrix addition and subtraction. Perform the indicated operation, if possible. (1) 5F(4) (1) (0) 5F8 0 () (7)G 4 14G a. F4 1 1 0 7G 8 b. 4F 5 0G 1F 3 8 4() 4(8) 6 5G5F 4(5) 4(0)G 1F 3 8 6 5G 8 3 5F 0 0G 1F 3 8 6 5G 8 1 (3) 3 1 8 5F 0 1 6 01 (5)G 11 4 5F 6 5G GUIDED PRACTICE for Examples 1 and Perform the indicated operation, if possible. 5 11 3 1 5 1. F 4 6 8G 1F. 8 4G F4 0 7 3 1GF 3 0 5 14G 1 3 3. 4F7 6 1 4. 0 5G 3F 4 1 3 5G 1F 0 6G MATRIX PROPERTIES Many of the properties you have used with real numbers can be applied to matrices as well. CONCEPT SUMMARY For Your Notebook Properties of Matrix Operations Let A, B, and C be matrices with the same dimensions, and let k be a scalar. Associative Property of Addition (A 1 B) 1 C 5 A 1 (B 1 C) Commutative Property of Addition Distributive Property of Addition Distributive Property of Subtraction A 1 B 5 B 1 A k(a 1 B) 5 ka 1 kb k(a B) 5 ka kb 188 Chapter 3 Linear Systems and Matrices

ORGANIZING DATA Matrices are useful for organizing data and for performing the same operations on large numbers of data values. E XAMPLE 3 TAKS REASONING: Multi-Step Problem MANUFACTURING A company manufactures small and large steel DVD racks with wooden bases. Each size of rack is available in three types of wood: walnut, pine, and cherry. Sales of the racks for last month and this month are shown below. Small Rack Sales Walnut Pine Cherry Large Rack Sales Walnut Pine Cherry Last month This month 15 78 5 95 316 05 Last month This month 100 51 70 114 15 300 Organize the data using two matrices, one for last month s sales and one for this month s sales. Then write and interpret a matrix giving the average monthly sales for the two month period. Solution STEP 1 Organize the data using two 3 3 matrices, as shown. Walnut Pine Cherry Last Month (A) This Month (B) Small Large Small Large F15 100 78 51 5 70G F95 114 316 15 05 300G ANOTHER WAY You can also evaluate } 1 (A 1 B) by first using the distributive property to rewrite the expression as } 1 A 1 } 1 B. STEP STEP 3 Write a matrix for the average monthly sales by first adding A and B to find the total sales and then multipling the result by } 1. 100 95 114 1 }( A 1 B) 5 } 1 78 51 316 15 SF15 5 70G1F05 300GD 5 1 } F0 14 594 466 430 570G 5F110 107 97 33 15 85G Interpret the matrix from Step. The company sold an average of 110 small walnut racks, 107 large walnut racks, 97 small pine racks, 33 large pine racks, 15 small cherry racks, and 85 large cherry racks. 3.5 Perform Basic Matrix Operations 189

SOLVING MATRIX EQUATIONS You can use what you know about matrix operations and matrix equality to solve an equation involving matrices. E XAMPLE 4 Solve a matrix equation 5x 3SF 6 4G 1F 3 7 Solve the matrix equation for x and y. Solution Simplify the left side of the equation. 5x 3SF 6 4G 1F 3 7 1 15 5 ygd 5F 1 15 5 ygd 5F 3 4G Write original equation. 3 4G 3F 5x 1 3 5 1 15 1 4 yg 5F Add matrices inside parentheses. 3 4G F15x 1 9 15 1 15 3 1 3yG 5F Perform scalar multiplication. 3 4G Equate corresponding elements and solve the two resulting equations. 15x 1 9 5 1 1 3y 5 4 x 5 y 5 4 c The solution is x 5 and y 5 4. GUIDED PRACTICE for Examples 3 and 4 5. In Example 3, find B A and explain what information this matrix gives. 3x 1 6. Solve SF 4 yg 1F 9 4 1 10 5 3GD 5F 18G for x and y. 3.5 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 5, 1, and 33 5 TAKS PRACTICE AND REASONING Exs. 8, 9, 33, 34, 36, and 37 1. VOCABULARY Copy and complete: The? of a matrix with 3 rows and 4 columns are 3 3 4.. WRITING Descri be how to determine whether two matrices are equal. EXAMPLE 1 on p. 187 for Exs. 3 9 3. ERROR ANALYSIS Describe and correct the error in adding the matrices. F5G 1F 3.8G 5F 9 4.1 9 4.1 5 3.8G 190 Chapter 3 Linear Systems and Matrices

ADDING AND SUBTRACTING MATRICES Perform the indicated operation, if possible. If not possible, state the reason. 4. F 5 8 10 10 8 1 3 1 8G 1F 5. 6 3G F 5 3G F 7. F1. 5.3 0.6 0.1 4.4 6.1 3.1 8. 6. 0.7G1F.4 8.1 1.9G G F8 3 1 0 9 1 6 3 9. 4 5G1F5 8 1 1G 3 4G 6. F 4 5 8 1G F F 7 3 9 1 5 6 4 11GF 5G 6 EXAMPLE on p. 188 for Exs. 10 15 MULTIPLYING BY A SCALAR Perform the indicated operation. 10. F 1 4 3 6G 11. 3F 0 5 4 7 3G 1. 4F 3 5 } 8 11 3.4 1.6 13. 1.5F 14. 5.4 0 3G 8 1 1 } 0 1 0 F G 15. 8 10 MATRIX OPERATIONS Use matrices A, B, C, and D to evaluate the matrix expression. 18 1 1.8 1.5 10.6 B 3 1G 5F C 6 0G 5F D 8.8 3.4 0G 5F A 5F 5 4 16. A 1 B 17. B A 18. 4A B 19. } 7 }4G.F 6 3.1 4.5 1 0.5 5.5 1.8 6.4G 7. 0 5.4.1 1.9 3.3G } 3 B 0. C 1 D 1. C 1 3D. D C 3. 0.5C D EXAMPLE 4 on p. 190 for Exs. 4 7 SOLVING MATRIX EQUATIONS Solve the matrix equation for x and y. 1 3x 1 18 4. F 4 5G 5F y 5G 5. F x 6 1 8G 1 F 5 1 7 6G 5F 9 4 6. F 8 x 5 6G F 3 9 10 4yG 5F 13 4 0 16G 7. 4xF 1 3 6G 5F 8 16 4 3yG 13 yg 8. TAKS REASONING Based on the equation below, what is the value of the expression 3x y? F x 0 0.5 0.75G 5F 6.4 0 0.5 3yG A 7.15 B 9.1 C 10.1 D 0.7 9. Find two matrices A and B such that A 3B 5F 5 0 TAKS REASONING 30. CHALLENGE Find the matrix X that makes the equation true. a. X 1F 5 0 4 3G 5F 7 8 3 5G b. X F 3 5 0G 5F 8 6 c. X 1F 3 1 4 7G 5F 8 9 11 6 d. 3X 0 10G F 1G 5F 1 G. 1 3G 13 15 19 G 3.5 Perform Basic Matrix Operations 191

PROBLEM SOLVING EXAMPLE 3 on p. 189 for Exs. 31 34 31. SNOWBOARD SALES A sporting goods store sells snowboards in several different styles and lengths. The matrices below show the number of each type of snowboard sold in 003 and 004. Write a matrix giving the change in sales for each type of snowboard from 003 to 004. Freeride Alpine Freestyle Sales for 003 Sales for 004 150 cm 155 cm 160 cm 165 cm 150 cm 155 cm 160 cm 165 cm F3 4 9 0 1 17 5 16 8 40 3 1G F3 47 30 19 5 16 0 14 9 39 36 31G 3. FUEL ECONOMY A car dealership sells four different models of cars. The fuel economy (in miles per gallon) is shown below for each model. Organize the data using a matrix. Then write a new matrix giving the fuel economy figures for next year s models if each measure of fuel economy increases by 8%. Economy car: Mid-size car: Mini-van: SUV: 3 mpg in city driving, 40 mpg in highway driving 4 mpg in city driving, 34 mpg in highway driving 18 mpg in city driving, 5 mpg in highway driving 19 mpg in city driving, mpg in highway driving 33. TAKS REASONING In a certain city, an electronics chain has a downtown store and a store in the mall. Each store carries three models of digital camera. Sales of the cameras for May and June are shown. May Downtown sales: 31 of model A, 4 of model B, 18 of model C Mall sales: of model A, 5 of model B, 11 of model C June Downtown sales: 5 of model A, 36 of model B, 1 of model C Mall sales: 38 of model A, 3 of model B, 15 of model C a. Organize the information using two matrices M and J that represent the sales for May and June, respectively. b. Find M 1 J and describe what this matrix sum represents. c. Write a matrix giving the average monthly sales for the two month period. 34. TAKS REASONING The matrices below show the numbers of female athletes who participated in selected NCAA sports and the average team size for each sport during the 000 001 and 001 00 seasons. Does the matrix A 1 B give meaningful information? Explain. Basketball Gymnastics Skiing Soccer 000 001 (A) 001 00 (B) Athletes Team size Athletes Team size 314,439 14.5 1,397 15.7 56 11.9 18,548.54 Basketball Gymnastics Skiing Soccer 314,54 14.3 1,440 16. 496 11.0 19,467.44 19 5 WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING

35. CHALLENGE A rectangle has vertices (1, 1), (1, 4), (5, 1), and (5, 4). Write a 3 4 matrix A whose columns are the vertices of the rectangle. Multiply matrix A by 3. In the same coordinate plane, draw the rectangles represented by the matrices A and 3A. How are the rectangles related? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Skills Review Handbook p. 1004; TAKS Workbook 36. TAKS PRACTICE A health teacher surveyed 100 students to determine their favorite exercise activity or combination of exercise activities. The results are shown at the right. How many of the students surveyed chose only running as their favorite exercise activity? TAKS Obj. 10 A 13 B 9 C 4 D 71 running 15 bicycling 4 6 18 8 3 8 swimming REVIEW Lesson.3; TAKS Workbook 37. TAKS PRACTICE Which statement best describes the effect on the graph shown when the y-intercept is decreased by 4? TAKS Obj. 3 F The x-intercept decreases. G The slope decreases. H The x-intercept increases. J The slope increases. 4 3 1 431 1 1 4 x 3 4 y QUIZ for Lessons 3.3 3.5 Graph the system of inequalities. (p. 168) 1. y < 6. x 1 3. x 1 3y > 3 x 1 y > x 1 y 5 x 1 3y < 9 4. x y 4 5. x 1 y 10 6. y < x x 1 4y 10 y x 1 y < 5x 1 9 Solve the system using any algebraic method. (p. 178) 7. x y 3z 5 5 8. x 1 y 1 z 53 9. x 4y 1 3z 5 1 x 1 y 5z 511 x 3y 1 z 5 9 6x 1 y 1 10z 5 19 x 3y 5 10 4x 5y 1 z 5 16 x 1 5y z 5 Use matrices A, B, and C to evaluate the matrix expression, if possible. If not possible, state the reason. (p. 187) A 5F 5 3 1G B 5F 4 3 6 9 C 8 10G 5F 1 4 1G 10. A 1 B 11. B A 1. 3A 1 C 13. 14. APPLES You have $5 to spend on 1 pounds of three types of apples. Empire apples cost $1.40 per pound, Red Delicious apples cost $1.10 per pound, and Golden Delicious apples cost $1.30 per pound. You want twice as many Red Delicious apples as the other two kinds combined. Use a system of equations to find how many pounds of each type you should buy. (p. 178) } 3 C EXTRA PRACTICE for Lesson 3.5, p. 101 ONLINE QUIZ at classzone.com 193

Graphing Calculator ACTIVITY Use after Lesson 3.5 3.5 Use Matrix Operations TEKS a.1, a.5, a.6 TEXAS classzone.com Keystrokes QUESTION How can you use a graphing calculator to perform matrix operations? E XAMPLE Perform operations with matrices Using matrices A and B below, find A 1 B and 3A B. A 5F 8 1 B 3 7 9G 5F 1 0 5 4 6 10G STEP 1 Enter matrix A Enter the dimensions and elements of matrix A. STEP Enter matrix B Enter the dimensions and elements of matrix B. STEP 3 Perform calculations From the home screen, calculate A 1 B and 3A B. MATRIX[A] 33 [8-1 ] [3-7 9],3=9 MATRIX[B] 33 [1 0-5] [-4 6 10],3=10 [A]+[B] [[9-1 -3] [-1-1 19]] 3[A]-[B] [[ -3 16] [17-33 7 ]] PRACTICE Use a graphing calculator to perform the indicated operation(s). 1. F 7 3 1 8 1.4 6.8 1. 5 G 1F. 3 6G.6F 0.8 5.6 3.G F 3 1 10 3 3. 1 5 6 0 6 1 4. 4 13 0G1F9 14 7 8G 3F 4 3 8 8 7 7 9 1 GF5 4 3G 5. BOOK SALES The matrices below show book sales (in thousands of dollars) at a chain of bookstores for July and August. The book formats are hardcover and paperback. The categories of books are romance (R), mystery (M), science fiction (S), and children s (C). Find the total sales of each format and category for July and August. July August R M S C R M S C Hardcover 18 16 1 13 6 0 17 8 Paperback F36 0 14 30G F40 4 8 0G 194 Chapter 3 Linear Systems and Matrices

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback