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
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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
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