TEKS (3) The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions.

TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Additional TEKS (1)(A)

TEKS FOCUS

•Corresponding elements – elements in the same position in each matrix

•Equal matrices – Equal matrices have the same dimensions and equal corresponding elements.

•Matrix equation – an equation in which the variable is a matrix

•Zero matrix – The zero matrix O, or Om*n, is the m * n matrix whose elements are all zeroes.

•Analyze – closely examine objects, ideas, or relationships to learn more about their nature

VOCABULARY

You can extend the addition and subtraction of numbers to matrices.

ESSENTIAL UNDERSTANDING

To add matrices A and B with the same dimensions, add corresponding elements. Similarly, to subtract matrices A and B with the same dimensions, subtract corresponding elements.

A = ca11 a12a21 a22

d B = cb11 b12b21 b22

d

A + B = ca11 + b11 a12 + b12a21 + b21 a22 + b22

d A - B = ca11 - b11 a12 - b12a21 - b21 a22 - b22

d

Key Concept Matrix Addition and Subtraction

If A, B, and C are m * n matrices, then

Example PropertyA + B is an m * n matrix Closure Property of AdditionA + B = B + A Commutative Property of Addition(A + B) + C = A + (B + C) Associative Property of AdditionThere is a unique m * n matrix

Additive Identity Property

O such that O + A = A + O = AFor each A, there is a unique

Additive Inverse Property

opposite, - A, such that A + (-A) = O

Properties Properties of Matrix Addition

4-1 Adding and Subtracting Matrices

116 Lesson 4-1 Adding and Subtracting Matrices

Problem 2

Problem 1

Adding and Subtracting Matrices

Given C ∙ c 3 2 4∙1 4 0

d and D ∙ c 1 4 3∙2 2 4

d , what are the following?

A C ∙ D B C ∙ D

c 3 2 4

-1 4 0d + c 1 4 3

-2 2 4d c 3 2 4

-1 4 0d - c 1 4 3

-2 2 4d

= c 3 + 1 2 + 4 4 + 3

-1 + (-2) 4 + 2 0 + 4d = c 3 - 1 2 - 4 4 - 3

-1 - (-2) 4 - 2 0 - 4d

= c 4 6 7

-3 6 4d = c 2 -2 1

1 2 -4d

Solving a Matrix Equation

Sports The first table shows the teams with the four best records halfway through their season. The second table shows the full season records for the same four teams. Which team had the best record during the second half of the season?

•Usetheequation:firsthalfrecords+ secondhalfrecords = seasonrecords.

•Solvethematrixequation.

Recordsforthesecondhalfoftheseason

•Recordsforthefirsthalfoftheseason

•Recordsforthefullseason

Step 1 Write 4 * 2 matrices to show the information from the two tables.

Let A = the first half records

B = the second half records A = ≥30 11

29 12

25 16

24 17

¥ F = ≥53 29

67 15

58 24

61 21

¥ F = the final records

TEKS Process Standard (1)(A)

Records for the First Half of the Season

Team 1

Team 2

Team 3

Team 4

Team

30

29

25

24

Wins

11

12

16

17

Losses

Records for Season

Team 1

Team 2

Team 3

Team 4

Team

53

67

58

61

Wins

29

15

24

21

Losses

continued on next page ▶

To add matrices they need to have the same dimensions. What are the dimensions of C ? Chas2rowsand3columns,soit’sa2 * 3 matrix.

117PearsonTEXAS.com

Problem 4

Problem 3

continuedProblem 2

Step 2 Solve A + B = F for B.

B = F - A

B = ≥53 29

67 15

58 24

61 21

¥ - ≥30 11

29 12

25 16

24 17

¥ = ≥53 - 30 29 - 11

67 - 29 15 - 12

58 - 25 24 - 16

61 - 24 21 - 17

¥ = ≥23 18

38 3

33 8

37 4

¥

Team 2 had the best record (38 wins and 3 losses) during the second half of the season.

Using Identity and Opposite Matrices

What are the following sums?

A c1 25 ∙7

d ∙ c0 00 0

d B c 2 8∙3 0

d ∙ c ∙2 ∙83 0

d

∙ c1 ∙ 0 2 ∙ 05 ∙ 0 ∙7 ∙ 0

d ∙ c1 25 ∙7

d ∙ c2 ∙ (∙2) 8 ∙ (∙8)∙3 ∙ 3 0 ∙ 0

d ∙ c0 00 0

d

TEKS Process Standard (1)(F)

Finding Unknown Matrix Values

Multiple Choice What values of x and y make the equation true?

c 9 3x + 1

2y - 1 10d = c 9 16

-5 10d

x = 3, y = 5 x = 5, y = -2

x = 173 , y = 5 x = 5, y = -3

3x + 1 = 16 Setcorrespondingelementsequal. 2y - 1 = -5

3x = 16 - 1 Isolatethevariableterm. 2y = -5 + 1

3x = 15 Simplify. 2y = -4

x = 5 Solveforxandy. y = -2

The correct answer is C.

How is this like adding real numbers?Addingzeroleavesthe matrixunchanged.Addingoppositesgiveyouzero.

What are the dimensions of matrix B? B willhave4rowsand2columns.Itisa4 * 2matrix.

How can you solve the equation?Forthetwomatricestobeequal,thecorrespondingelementsmustbeequal.

118 Lesson 4-1 Adding and Subtracting Matrices

PRACTICE and APPLICATION EXERCISESON

LINE

HO

M E W O RK

For additional support whencompleting your homework, go to PearsonTEXAS.com.

1. Apply Mathematics (1)(A) The table shows the number of beach balls produced during one shift at two manufacturing plants. Plant 1 has two shifts per day and Plant 2 has three shifts per day. Write matrices to represent one day’s total output at each plant. Find the difference in daily production totals at the two plants.

Find each matrix sum or difference if possible. If not possible, explain why.

A = £3 46 −21 0

§ B = £−3 1

2 −4−1 5

§ C = c 1 2−3 1

d D = c5 10 2

d

2. A + B 3. B + D 4. B - A 5. C - D

6. Use Representations to Communicate Mathematical Ideas (1)(E) The modern pentathlon is a grueling all-day competition. Each member of a team competes in five events: target shooting, fencing, swimming, horseback riding, and cross-country running. Here are scores for the U.S. women at the 2004 Olympic Games.

a. Write two 5 * 1 matrices to represent each woman’s scores for each event.

b. Find the total score for each athlete.

Find each sum.

7. c 2 -3 4

5 6 -7d + c 0 0 0

0 0 0d 8. c 6 -3

-7 2d + c -6 3

7 -2d

Find the value of each variable.

9. c 2 2

-1 6d - c 4 -1

0 5d = c x y

-1 zd 10. c 2 4

8 4.5d = c 4x - 6 -10t + 5

4x 15t + 1.5xd

Solve each matrix equation.

11. £1 2

2 1

-3 4

§ + X = £5 -6

1 0

8 5

§ 12. c 2 1 -1

0 2 1d - X = c 11 3 -13

15 -9 8d

13. X - c 1 4

-2 3d = c 5 -2

1 0d 14. X + c 6 1

-2 3d = c 2 0

-3 1d

Beach Ball Production Per Shift

3-color1-color

Plastic500

400

Rubber700

1200

Plastic1300

600

Rubber1900

1600

Plant 1

Plant 2

U.S. Women’s Pentathlon Scores, 2004 Olympics

Shooting

Fencing

Swimming

Riding

Running

Event

952

720

1108

1172

1044

760

832

1252

1144

1064

Anita Allen Mary Beth lagorashvili

SOURCE: Athens 2004 Olympic Games

Scan page for a Virtual Nerd™ tutorial video.

119PearsonTEXAS.com

TEXAS Test Practice

22. What is the sum c 5 7 3

-1 0 -4d + c -7 4 2

1 -2 -3d ?

A. c -2 11 5

0 -2 -7d C. c 12 3 1

-2 2 -1d

B. c -35 28 6

-1 0 12d D. The matrices cannot be added.

23. Which arithmetic sequence includes the term 27?

I. a(1) = 7, a(n) = a(n - 1) + 5

II. a(n) = 3 + 4(n - 1)

III. a(n) = 57 - 6n

F. I only G. I and II only H. II and III only J. I, II, and III

15. Use Representations to Communicate Mathematical Ideas (1)(E) Refer to the table at the right.

a. Add two matrices to find the total number of people participating in each activity.

b. Subtract two matrices to find the difference between the numbers of males and females in each activity.

c. In part (b), does the order of the matrices matter? Explain.

16. Analyze Mathematical Ideas (1)(F) Given a matrix A, explain how to find a matrix B such that A + B = 0.

Solve each equation for each variable.

17. C4b + 2 -3 4d-4a 2 3

2f - 1 -14 1

S = C 11 2c - 1 0

-8 2 3

0 3g - 2 1

S 18. C 4c 2 - d 5

-3 -1 2

0 -10 15

S = C2c + 5 4d g-3 h f - g

0 -4c 15

S 19. Find the sum of E = £

3

4

7

§ and the additive inverse of G = £-2

0

5

§ .

20. Prove that matrix addition is commutative for 2 * 2 matrices.

21. Prove that matrix addition is associative for 2 * 2 matrices.

U.S. Participation (millions) inSelected Leisure Activities

Movies

Exercise Programs

Sports Events

Home Improvement

Activity

59.2

54.3

40.5

45.4

Male

65.4

59.0

31.1

41.8

Female

SOURCE: U.S. National Endowment for the Arts

120 Lesson 4-1 Adding and Subtracting Matrices

Technology Lab Working With Matrices

Use With Lesson 4-1 Foundational to teks (3)(B), (1)(E)

You can use a graphing calculator to work with matrices. First you need to know how to enter a matrix into the calculator.

Example 1

Enter matrix A ∙ £∙3 4

7 ∙50 ∙2

§ into your graphing calculator.

Select the EDIT option of the matrix feature to edit matrix [A].

Specify a 3 * 2 matrix by pressing 3 enter 2 enter .

Enter the matrix elements one row at a time, pressing enter after each element.

Then use the quit feature to return to the main screen.

1: [A]2: [B]3: [C]4: [D]5: [E]

NAMES MATH EDIT

MATRIX [A] 3 �2[ 0[ 0[ 0

1, 1 � 0

000

]]]

MATRIX [A] 3 �2[ �3[ 7[ 0

3, 2 � �2

4�5�2

]]]

continued on next page ▶

121PearsonTEXAS.com

Technology Lab continued

ExercisesFind each sum or difference.

1. c 0 -3

5 -7d - c -5 3

4 10d

2. c 3 5 -7

0 -2 0d - c -1 6 2

-9 4 0d

3. c 3

5d - c -6

7d

4. [3 5 -8] + [-6 4 1]

5. c 17 8 0

3 -5 2d - c 4 6 5

2 -2 9d

6. [-9 6 4] + [-3 8 4]

Example 2

Given A = £−3 4

7 −50 −2

§ and B = £10 −7

4 −3−12 11

§ , find A + B and A − B.

Enter both matrices into the calculator. Use the names option of the matrix feature to select each matrix. Press enter to see the sum.

Repeat the corresponding steps to find the difference A - B.

[A] � [B] [ [ 7 �3 [ 11 �8 [ �12 9

]]] ]

122 Technology Lab Working With Matrices