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F. Solving Applications Using Matrices

# F. Solving Applications Using Matrices

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CHAPTER 7 Matrices and Matrix Applications

EXAMPLE 9

Determining the Original Value of Collector’s Items

A museum purchases a famous painting, a ruby tiara, and a rare coin for its collection, spending

a total of \$30,000. One year later, the painting has tripled in value, while the tiara and the coin

have doubled in value. The items now have a total value of \$75,000. Find the purchase price of

each if the original price of the painting was \$1000 more than twice the coin.

Solution

Let P represent the price of the painting, T the tiara, and C the coin.

Total spent was \$30,000:

One year later:

Value of painting versus coin:

S P ϩ T ϩ C ϭ 30,000

S 3P ϩ 2T ϩ 2C ϭ 75,000

S P ϭ 2C ϩ 1000

P ϩ T ϩ C ϭ 30,000

1P ϩ 1T ϩ 1C ϭ 30,000

• 3P ϩ 2T ϩ 2C ϭ 75,000 standard form S • 3P ϩ 2T ϩ 2C ϭ 75,000

P ϭ 2C ϩ 1000

1P ϩ 0T Ϫ 2C ϭ 1000

1

£3

1

1

2

0

1

£0

0

1

1

1

1

2

Ϫ2

30000

75000 §

1000

1

1

3

30000

15000 §

29000

Ϫ3R1 ϩ R2 S R2

Ϫ1R1 ϩ R3 S R3

Ϫ1R2 ϩ R3 S R3

1

£0

0

1

Ϫ1

Ϫ1

1

£0

0

1

1

0

1

Ϫ1

Ϫ3

1

1

2

30000

Ϫ15000 §

Ϫ29000

30000

15000 §

14000

matrix form S

1

£3

1

Ϫ1R2 S R2

Ϫ1R3 S R3

R3

S R3

2

1

2

0

1

2

Ϫ2

30000

75000 §

1000

1

£0

0

1

1

1

1

1

3

30000

15000 §

29000

1

£0

0

1

1

0

1

1

1

30000

15000 §

7000

From R3 of the triangularized form, C ϭ \$7000 directly. Since

R2 represents T ϩ C ϭ 15,000, we find the tiara was purchased

for T ϭ \$8000. Substituting these values into the first equation

shows the painting was purchased for \$15,000. The solution is

(15,000, 8000, 7000). The solution found using a graphing

calculator is also shown.

F. You’ve just seen how

we can solve applications

using matrix methods

Now try Exercises 61 through 68 ᮣ

7.1 EXERCISES

CONCEPTS AND VOCABULARY

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.

1. A matrix with the same number of rows and

columns is called a(n)

matrix.

2

4

Ϫ3

d is a

by

1 Ϫ2

1

matrix. The entry in the second row and third

column is

.

3. Matrix A ϭ c

5. The notation Ϫ2R1 ϩ R2 S R2 indicates that an

equivalent matrix is formed by performing what

operations/replacements?

2. When the coefficient matrix is used with the matrix

of constants, the result is a(n)

matrix.

4. Given matrix B shown

here, the diagonal entries

are

,

, and

.

1

4

3

B ϭ £ Ϫ1

5

3

Ϫ2 1

6. Describe how to tell an inconsistent system apart

from a dependent system when solving using

matrix methods (row reduction).

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647

Determine the size of each matrix and identify the third

row and second column entry. If the matrix given is a

square matrix, identify the diagonal entries.

1

7. Ê 2.1

3

1

1

9.

5

2

0

1 Đ

5.8

0

3

1

3

1

8. Ê 1

5

0

3

1

4

7 Đ

2

4

7

Ơ

2

9

Form the augmented matrix, then name the diagonal

entries of the coefficient matrix.

2x Ϫ 3y Ϫ 2z ϭ 7

10. • x Ϫ y ϩ 2z ϭ Ϫ5

3x ϩ 2y Ϫ z ϭ 11

x ϩ 2y Ϫ z ϭ 1

zϭ3

11. • x ϩ

2x Ϫ y ϩ z ϭ 3

2x ϩ 3y ϩ z ϭ 5

12. •

2y Ϫ z ϭ 7

x Ϫ y Ϫ 2z ϭ 5

Write the system of equations for each matrix. Then use

back-substitution to find its solution.

13. c

1

0

4

1

14. c

1

0

Ϫ5

Ϫ1

5

1

2

d

Ϫ15

d

Ϫ2

1

15. £ 0

0

2

1

0

Ϫ1

2

1

0

3

1

16. £ 0

0

0

1

0

7

Ϫ5

1

Ϫ5

15 §

Ϫ26

1

17. £ 0

0

3

1

0

Ϫ4

Ϫ32

1

29

1

18. £ 0

0

2

1

0

Ϫ1

3

1

6

1

21

2

3

§

2

3 §

22

7

Perform the indicated row operation(s) in the order

given and write the new matrix.

Ϫ3

2

1

2

19. c

Ϫ5

20. c

7

4

Ϫ1

d

4

1

4 R2

3

d

12

4

Ϫ8

2R1 S R1,

5R1 ϩ R2 S R2

S R2,

R1 4 R2

Ϫ2

21. £ 5

1

1

8

Ϫ3

Ϫ3

22. £ 1

4

2

1

1

0

2

Ϫ3

0

2

3

23. £ 6

4

1

Ϫ1

Ϫ2

1

Ϫ1

Ϫ3

8

10 §

22

2

24. £ 3

4

1

1

3

0

3

3

Ϫ1

1

2

4

Ϫ5 §

2

Ϫ3

0 §

3

R1 4 R3,

Ϫ5R1 ϩ R2 S R2

R1 4 R2,

Ϫ4R1 ϩ R3 S R3

Ϫ2R1 ϩ R2 S R2,

Ϫ4R1 ϩ 3R3 S R3

Ϫ3R1 ϩ 2R2 S R2,

Ϫ2R1 ϩ R3 S R3

What row operations would produce zeroes beneath the

first entry in the diagonal?

1

25. £ Ϫ2

3

3

4

Ϫ1

0

1

Ϫ2

2

9

1

26. £ 3

Ϫ5

1

0

3

Ϫ4

1

2

Ϫ3

5 §

3

1

27. £ 5

Ϫ4

2

1

3

0

2

Ϫ3

10

6 §

2

Solve each system by triangularizing the augmented

matrix and using back-substitution. Simplify by

clearing fractions or decimals before beginning.

28. e

2y ϭ 5x ϩ 4

Ϫ5x ϭ 2 Ϫ 4y

29. e

0.15g Ϫ 0.35h ϭ Ϫ0.5

Ϫ0.12g ϩ 0.25h ϭ 0.1

Ϫ1u ϩ 14v ϭ 1

30. e 15

1

10 u ϩ 2 v ϭ 7

2x Ϫ 3y Ϫ 2z ϭ 7

32. • x Ϫ y ϩ 2z ϭ Ϫ5

3x ϩ 2y Ϫ z ϭ 11

x Ϫ 2y ϩ 2z ϭ 7

31. • 2x ϩ 2y Ϫ z ϭ 5

3x Ϫ y ϩ z ϭ 6

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Solve using Gauss-Jordan elimination.

x ϩ 2y Ϫ z ϭ 1

33. • x ϩ

zϭ3

2x Ϫ y ϩ z ϭ 3

2x ϩ 3y ϩ z ϭ 5

34. •

2y Ϫ z ϭ 7

x Ϫ y Ϫ 2z ϭ 5

Ϫx ϩ y ϩ 2z ϭ 2

x ϩ y Ϫ 2z ϭ Ϫ1

35. • x ϩ y Ϫ z ϭ 1 36. • 4x Ϫ y ϩ 3z ϭ 3

2x ϩ y ϩ z ϭ 4

3x ϩ 2y Ϫ z ϭ 4

Solve each system of equations using a graphing

calculator. Verify each solution on the home screen using

the ALPHA keys, as in Chapter 6.

x ϩ 2y ϩ 3z ϭ 9

37. • Ϫ2x ϩ y Ϫ 5z ϭ Ϫ20

3x Ϫ 5y ϩ z ϭ 14

3x ϩ 5y ϩ z ϭ 6

38. • Ϫ4x ϩ 7y Ϫ 2z ϭ 24

x Ϫ 10y ϩ 3z ϭ Ϫ40

0.2x ϩ 0.5y ϩ

z ϭ 13

39. • Ϫx ϩ 0.7y Ϫ 0.4z ϭ Ϫ9

Ϫ0.5x Ϫ

y ϩ 0.8z ϭ Ϫ4.6

0.1x ϩ 2y ϩ 0.6z ϭ Ϫ36.4

40. • Ϫ3x ϩ 0.8y Ϫ 0.2z ϭ Ϫ35

0.5x Ϫ

y ϩ 0.4z ϭ 25

ϩ 34y ϩ z ϭ 34

41. • Ϫx ϩ 14y ϩ 32z ϭ Ϫ58

1

1

5

4 x Ϫ y ϩ 2 z ϭ Ϫ2

1

2x

ϩ 12y ϩ z ϭ Ϫ14

42. • Ϫx ϩ 32y Ϫ 78z ϭ Ϫ35

1

2

1

4x Ϫ y ϩ 4z ϭ 5

5

4x

Solve each system by triangularizing the augmented

matrix and using back-substitution. If the system is

linearly dependent, give the solution in terms of a

parameter. If the system has coincident dependence,

answer in set notation as in Chapter 6.

4x Ϫ 8y ϩ 8z ϭ 24

43. • 2x Ϫ 6y ϩ 3z ϭ 13

3x ϩ 4y Ϫ z ϭ Ϫ11

3x ϩ y ϩ z ϭ Ϫ2

44. • x Ϫ 2y ϩ 3z ϭ 1

2x Ϫ 3y ϩ 5z ϭ 3

x ϩ 3y ϩ 5z ϭ 20

45. • 2x ϩ 3y ϩ 4z ϭ 16

x ϩ 2y ϩ 3z ϭ 12

Ϫx ϩ 2y ϩ 3z ϭ Ϫ6

x Ϫ y ϩ 2z ϭ Ϫ4

3x Ϫ 6y Ϫ 9z ϭ 18

3x Ϫ 4y ϩ 2z ϭ Ϫ2

47. • 32x Ϫ 2y ϩ z ϭ Ϫ1

Ϫ6x ϩ 8y Ϫ 4z ϭ 4

46. •

2x Ϫ y ϩ 3z ϭ 1

48. • 4x Ϫ 2y ϩ 6z ϭ 2

10x Ϫ 5y ϩ 15z ϭ 5

49. •

2x Ϫ y ϩ 3z ϭ 1

2y ϩ 6z ϭ 2

x Ϫ 12y ϩ 32z ϭ 5

x Ϫ 2y ϩ 3z ϭ 2

50. • 3x ϩ 4y Ϫ z ϭ 6

4x ϩ 2y ϩ 2z ϭ 7

x ϩ 2y ϩ z ϭ 4

51. • 3x Ϫ 4y ϩ z ϭ 4

6x Ϫ 8y ϩ 2z ϭ 8

Ϫ2x ϩ 4y Ϫ 3z ϭ 4

52. • 5x Ϫ 6y ϩ 7z ϭ Ϫ12

x ϩ 2y ϩ z ϭ Ϫ4

Solve each system of equations using a graphing

calculator. Verify each solution on the home screen using

the ALPHA keys, as in Chapter 6.

2x Ϫ 3y ϩ z ϭ 0

x ϩ 2y Ϫ z ϭ 5

53. μ

3x ϩ

2z ϭ 4

x ϩ 3y ϩ z ϭ 4

Ϫ2x ϩ 5y

ϭ Ϫ4

3x Ϫ 4y ϩ z ϭ 0

54. μ

x ϩ y Ϫ 3z ϭ Ϫ8

5x Ϫ 9y ϩ z ϭ 4

x ϩ 3y ϩ z ϭ 1

2z ϭ 7

55. μ

Ϫ2y ϩ 3z ϭ Ϫ6

2x ϩ 3y Ϫ z ϭ 8

Ϫ2x ϩ 3y

ϭ4

Ϫ 2y ϩ 5z ϭ Ϫ19

56. μ

x

Ϫ 4z ϭ 13

Ϫx ϩ 3y Ϫ 4z ϭ 17

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x ϩ 2y Ϫ 3z ϩ w ϭ Ϫ13

Ϫ2x ϩ 3y ϩ z Ϫ 2w ϭ Ϫ3

57. μ

Ϫx Ϫ y ϩ 2z ϩ 3w ϭ 4

3x Ϫ 2y Ϫ z Ϫ w ϭ 5

649

Section 7.1 Solving Linear Systems Using Matrices and Row Operations

2x ϩ 3y Ϫ 4z ϩ w ϭ Ϫ10

x Ϫ 2y

Ϫ 3w ϭ 6

58. μ

3x

ϩ 5z ϩ 2w ϭ 1

Ϫx ϩ y ϩ 2z Ϫ 5w ϭ 21

WORKING WITH FORMULAS

1

Area of a triangle in the plane: A ‫( ؎ ؍‬x1 y2 ؊ x2 y1 ؉ x2 y3 ؊ x3 y2 ؉ x3 y1 ؊ x1 y3)

2

The area of a triangle in the plane is given by the formula shown, where the vertices of the triangle are located at the

points (x1, y1), (x2, y2), and (x3, y3), and the sign is chosen to ensure a positive value.

59. Find the area of a triangle whose vertices are

1Ϫ1, Ϫ32, (5, 2), and (1, 8).

60. Find the area of a triangle whose vertices are

16, Ϫ22, 1Ϫ5, 42 , and 1Ϫ1, 72.

APPLICATIONS

Model each problem using a system of linear equations. Then solve using the augmented matrix.

Descriptive Translation

61. The distance (via air travel) from Los Angeles (LA), California, to

Saint Louis (STL), Missouri, to Cincinnati (CIN), Ohio, to New York

City (NYC), New York, is approximately 2480 mi. Find the length of

each flight if the distance from LA to STL is 50 mi more than five

times the distance between STL and CIN, and 110 mi less than three

times the distance from CIN to NYC.

New York City

St. Louis

Cincinnati

Los Angeles

62. In the 2006 NBA Championship Series, Dwayne Wade of the Miami

Heat carried his team to the title after the first two games were lost to

the Dallas Mavericks. If 187 points were scored in the title game and the

Heat won by 3 points, what was the final score?

63. Moe is lecturing Larry and Curly once again (Moe, Larry, and Curly of The Three Stooges fame) claiming he is

twice as smart as Larry and three times as smart as Curly. If he is correct and the sum of their IQs is 165, what is

the IQ of each stooge?

64. A collector of rare books buys a handwritten,

autographed copy of Edgar Allan Poe’s Annabel

Lee, an original advance copy of L. Frank Baum’s

The Wonderful Wizard of Oz, and a first print copy

of The Caine Mutiny by Herman Wouk, paying a

total of \$100,000. Find the cost of each one, given

that the cost of Annabel Lee and twice the cost of

The Caine Mutiny sum to the price paid for The

Wonderful Wizard of Oz, and The Caine Mutiny

cost twice as much as Annabel Lee.

Geometry

65. A right triangle has a hypotenuse of 39 m. If the

perimeter is 90 m, and the longer leg is 6 m longer

than twice the shorter leg, find the dimensions of

the triangle.

66. In triangle ABC, the sum of angles A and C is equal

to three times angle B. Angle C is 10 degrees more

than twice angle B. Find the measure of each angle.

Investment and Finance

67. Suppose \$10,000 is invested in three different

investment vehicles paying 5%, 7%, and 9%

annual interest. Find the amount invested at each

rate if the interest earned after 1 yr is \$760 and the

amount invested at 9% is equal to the sum of the

amounts invested at 5% and 7%.

68. The trustee of a union’s pension fund has

invested funds in three ways: a savings fund

paying 4% annual interest, a money market fund

paying 7%, and government bonds paying 8%.

Find the amount invested in each if the interest

earned after one year is \$0.178 million and the

amount in government bonds is \$0.3 million

more than twice the amount in money market

funds. The total amount invested is \$2.5 million

dollars.

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CHAPTER 7 Matrices and Matrix Applications

EXTENDING THE CONCEPT

69. Given the drawing shown, use a system of

equations and the matrix method to find the

measure of the angles labeled as x and y. Recall

that vertical angles are equal and that the sum of

the angles in a triangle is 180°.

70. The system given here has a solution of 11, Ϫ2, 32.

Find the value of a and b.

1

£ 2b

2a

71Њ

a

2a

7

b

5

3b

1

13 §

Ϫ8

x

y

(x Ϫ 59)Њ

71. (3.5) Given f 1x2 ϭ x3 Ϫ 8 and g1x2 ϭ x Ϫ 2, find

f

f ϩ g, f Ϫ g, fg, and .

g

72. (4.2) Given x ϭ 2 is a zero of h1x2 ϭ x4 Ϫ x2 Ϫ 12,

find all zeroes of h, real and complex.

73. (5.6) Since 2005, cable installations for an Internet

company have been modeled by the function

C1t2 ϭ 15 ln 1t ϩ 12 , where C(t) represents cable

installations in thousands, t yr after 2005. In what

year will the number of installations be greater

than 30,000?

7.2

74. (5.6) If a set amount of money p is deposited

regularly (daily, weekly, monthly, etc.) n times per

year at a fixed interest rate r, the amount of money A

accumulated in t years is given by the formula

shown. If a parent deposited \$250 per month for

18 yr at 4.6% beginning when her first child was

born, how much has been accumulated to help pay

for college expenses?

r nt

p c a1 ϩ b Ϫ 1 d

n

r

n

The Algebra of Matrices

LEARNING OBJECTIVES

Matrices serve a much wider purpose than just a convenient method for solving systems. To understand their broader application, we need to know more about matrix

theory, the various ways matrices can be combined, and some of their more practical

uses. The common operations of addition, subtraction, multiplication, and division are

all defined for matrices, as are other operations. Practical applications of matrix theory

can be found in the social sciences, inventory management, genetics, operations

research, engineering, and many other fields.

matrices

C. Compute the product

of two matrices

A. Equality of Matrices

In Section 7.2 you will see

how we can:

A. Determine if two matrices

are equal

To effectively study matrix algebra, we first give matrices a more general definition.

For the general matrix A, all entries will be denoted using the lowercase letter “a,” with

their position in the matrix designated by the dual subscript aij. The letter “i” gives the

row and the letter “j ” gives the column of the entry’s location. The general m ϫ n

matrix A is written

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Section 7.2 The Algebra of Matrices

col 1

row 1 S

a11

row 2 S l a21

row 3 S

a31

row i S

row m S

p

col 2

a12

a22

a32

col 3

a13

a23

a33

p

p

p

col j

a1j

a2j

a3j

p

p

p

o

ai1

o

ai2

o

ai3

p

o

aij

p

am1

o

am2

o

am3

p

o

amj

p

z o

651

col n

a1n

a2n }

a3n

o p

ain

aij is a general

matrix element

o {

amn

The size of a matrix is also referred to as its order, and we say the order of general

matrix A is m ϫ n. Note that diagonal entries have the same row and column number,

aij, where i ϭ j. Also, where the general entry of matrix A is aij, the general entry of

matrix B is bij, of matrix C is cij, and so on.

EXAMPLE 1

Identifying the Order and Entries of a Matrix

State the order of each matrix and name the entries corresponding to a22, a31; b22,

b31; and c22, c31.

3

Ϫ2

0.2 Ϫ0.5 0.7

1

4

d

5 §

0.3

1 §

a. A ϭ c

b. B ϭ £ 1

c. C ϭ £ Ϫ1

Ϫ2 Ϫ3

Ϫ4

3

2.1 Ϫ0.1 0.6

Solution

a. matrix A: order 2 ϫ 2. Entry a22 ϭ Ϫ3 (the row 2, column 2 entry is Ϫ3).

There is no a31 entry (A is only 2 ϫ 2).

b. matrix B: order 3 ϫ 2. Entry b22 ϭ 5, entry b31 ϭ Ϫ4.

c. matrix C: order 3 ϫ 3. Entry c22 ϭ 0.3, entry c31 ϭ 2.1.

Now try Exercises 7 through 12

Equality of Matrices

Two matrices are equal if they have the same order and their corresponding entries are

equal. In symbols, this means that A ϭ B if aij ϭ bij for all i and j.

EXAMPLE 2

Determining If Two Matrices Are Equal

Determine whether the following statements are true, false, or conditional. If false,

explain why. If conditional, find values that will make the statement true.

3

Ϫ2

1

4

Ϫ3 Ϫ2

3 Ϫ2 1

a. c

b. £ 1

d ϭ c

d

5 § ϭ c

d

Ϫ2 Ϫ3

4

1

5 Ϫ4 3

Ϫ4

3

1

4

a Ϫ 2 2b

c. c

d ϭ c

d

Ϫ2 Ϫ3

c

Ϫ3

Solution

A. You’ve just seen how

we can determine if two

matrices are equal

Ϫ3 Ϫ2

1

4

d ϭ c

d is false. The matrices have the same order and

Ϫ2 Ϫ3

4

1

entries, but corresponding entries are not equal.

3

Ϫ2

3 Ϫ2 1

b. £ 1

5 § ϭ c

d is false. Their orders are not equal.

5 Ϫ4 3

Ϫ4

3

1

4

a Ϫ 2 2b

c. c

d ϭ c

d is conditional. The statement is true when

Ϫ2 Ϫ3

c

Ϫ3

a Ϫ 2 ϭ 1 1a ϭ 32, 2b ϭ 4 1b ϭ 22, c ϭ Ϫ2, and is false otherwise.

a. c

Now try Exercises 13 through 16

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CHAPTER 7 Matrices and Matrix Applications

B. Addition and Subtraction of Matrices

A sum or difference of matrices is found by combining the corresponding entries. This

limits the operations to matrices of like orders, so that every entry in one matrix has a

“corresponding entry” in the other. This also means the result is a new matrix of like

order, whose entries are the corresponding sums or differences. If you attempt to add

matrices of unlike size on a graphing calculator, an error message is displayed: ERR:

DIM MISMATCH. Note that since aij represents a general entry of matrix A, [aij] represents the entire matrix.

Given matrices A, B, and C having like orders.

The sum A ϩ B ϭ C,

where 3aij ϩ bij 4 ϭ 3cij 4.

EXAMPLE 3

The difference A Ϫ B ϭ C,

where 3aij Ϫ bij 4 ϭ 3cij 4.

Compute the sum or difference of the matrices indicated.

2

6

A ϭ £1

0 §

1 Ϫ3

a. A ϩ C

Solution

2

a. A ϩ C ϭ £ 1

1

Bϭ c

Ϫ3

Ϫ5

b. A ϩ B

6

3

0 § ϩ £ 1

Ϫ3

Ϫ4

2

4

3

Cϭ £ 1

Ϫ4

Ϫ1

d

3

Ϫ2

5 §

3

c. C Ϫ A

Ϫ2

5 §

3

sum of A and C

2ϩ3

6 ϩ 1Ϫ22

5

4

ϭ £ 1ϩ1

0 ϩ 5 §ϭ £ 2

1 ϩ 1Ϫ42

Ϫ3 ϩ 3

Ϫ3 0

2

6

Ϫ3 2 Ϫ1

b. A ϩ B ϭ £ 1

0 § ϩ c

d

Ϫ5 4

3

1 Ϫ3

3

Ϫ2

2

6

c. C Ϫ A ϭ £ 1

5 § Ϫ £1

0 §

Ϫ4

3

1 Ϫ3

3Ϫ2

Ϫ2 Ϫ 6

1

Ϫ8

ϭ £ 1Ϫ1

5Ϫ0 § ϭ £ 0

5 §

Ϫ4 Ϫ 1 3 Ϫ 1Ϫ32

Ϫ5

6

defined for matrices of unlike order.

difference of C and A

subtract corresponding entries

Now try Exercises 17 through 20

Operations on matrices can be very laborious for larger matrices and for matrices with

noninteger or large entries. For these, we can turn to available technology for

assistance. This shifts our focus from a meticulous computation of entries, to carefully

entering each matrix into the calculator, double-checking each entry, and appraising

results to see if they’re reasonable.

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Page 653

College Algebra G&M—

7–17

Section 7.2 The Algebra of Matrices

EXAMPLE 4

653

Using Technology for Matrix Operations

Use a calculator to compute the difference A Ϫ B for the matrices given.

2

11

A ϭ £ 0.9

0

Solution

Ϫ0.5

3

4

6

6

5

Ϫ4 §

5

Ϫ12

Bϭ £

Ϫ7

10

1

6

11

25

0

Ϫ4

Ϫ5

9

0.75

Ϫ5 §

Ϫ5

12

The entries for matrix A are shown in Figure 7.9. After entering matrix B, exit to

the home screen [ 2nd MODE (QUIT)], call up matrix A, press the ؊ (subtract)

key, then call up matrix B and press . The calculator quickly finds the difference

and displays the results shown in Figure 7.10. The last line on the screen shows the

result can be stored for future use in a new matrix C by pressing the STO key,

calling up matrix C, and pressing .

ENTER

ENTER

Figure 7.9

Figure 7.10

Now try Exercises 21 through 24

Figure 7.11

In Figure 7.10 the dots to the right on the calculator screen indicate there are additional digits or matrix columns that can’t fit on the display, as often happens with larger

matrices or decimal numbers. Sometimes, converting entries to fraction form will provide a display that’s easier to read. Here, this is done by calling up the matrix C, and

using the MATH 1: ᮣ Frac option. After pressing , all entries are converted to fractions in simplest form (where possible), as in Figure 7.11. The third column can be

viewed by pressing the right arrow.

Since the addition of two matrices is defined as the sum of corresponding entries,

we find the properties of matrix addition closely resemble those of real number addition. Similar to standard algebraic properties, ϪA represents the product Ϫ1 # A and

any subtraction can be rewritten as an algebraic sum: A Ϫ B ϭ A ϩ 1ϪB2. As noted in

the properties box, for any matrix A, the sum A ϩ 1ϪA2 will yield the zero matrix Z,

a matrix of like size whose entries are all zeroes. Also note that matrix ϪA is the

ENTER

Given matrices A, B, C, and Z are m ϫ n matrices, with Z the zero matrix. Then,

B. You’ve just seen how

matrices

I. A ϩ B ϭ B ϩ A

¡

II. 1A ϩ B2 ϩ C ϭ A ϩ 1B ϩ C2 S

III. A ϩ Z ϭ Z ϩ A ϭ A

¡

IV. A ϩ 1ϪA2 ϭ 1ϪA2 ϩ A ϭ Z ¡

ϪA is the additive inverse of A

C. Matrices and Multiplication

The algebraic terms 2a and ab have counterparts in matrix algebra. The product 2A

represents a constant times a matrix and is called scalar multiplication. The product

AB represents the product of two matrices.

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F. Solving Applications Using Matrices

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