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
2§
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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Section 7.1 Solving Linear Systems Using Matrices and Row Operations
647
DEVELOPING YOUR SKILLS
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
2§
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
6§
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
1§
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
xϪ
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
ᮣ
MAINTAINING YOUR SKILLS
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
Aϭ
r
n
The Algebra of Matrices
LEARNING OBJECTIVES
B. Add and subtract
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.
Addition and Subtraction of Matrices
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.
Adding and Subtracting Matrices
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
5§
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
add corresponding entries
Addition and subtraction are not
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
additive inverse for A, while Z is the additive identity.
ENTER
Properties of Matrix Addition
Given matrices A, B, C, and Z are m ϫ n matrices, with Z the zero matrix. Then,
B. You’ve just seen how
we can add and subtract
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 ¡
matrix addition is commutative
matrix addition is associative
Z is the additive identity
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.