5 Matrix Operations: Getting Information from Data
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612
CHAPTER 7
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Systems of Equations and Data in Categories
Solution
(a) Since A and B are matrices with the same dimension 13 * 22 , we can add
them.
2
A + B = £0
7
-3
1
5§ + £-3
- 12
2
0
3
1§ = £-3
2
9
0
6
d - c
5
8
-6
1
d = c
9
-8
-3
6§
3
2
(b) Since C and D are matrices with the same dimension 12 * 32 , we can
subtract them.
C -D = c
7
0
-3
1
0
1
-3
0
6
d
-4
(c) C + A is not defined because we cannot add matrices with different dimensions.
(d) To find 5A, we multiply each entry in A by 5.
2
5A = 5 £ 0
7
■
2
-3
10
5§ = £ 0
- 12
35
- 15
25 §
- 52
■
NOW TRY EXERCISES 7 AND 15
■ Matrix Multiplication
In Section 7.4 we learned how to multiply a matrix times a column matrix. We now
describe how to use this to define matrix multiplication in general (when the second
factor has more than just one column).
First, the product AB of two matrices A and B is defined only when the number
of columns in A is equal to the number of rows in B. This means that if we write their
dimensions side by side, the two inner numbers must match.
Matrices
Dimensions
A
m *n
B
n *k
Columns in A
Rows in B
If the dimensions of A and B match in this way, then the product AB is a matrix of
dimension m * k. Before describing the procedure for obtaining the elements of AB,
we define the inner product of a row of A and a column of B.
b1
b
If 3a1 a2 p an 4 is a row of A and if ≥ 2 ¥ is a column of B, then their
o
bn
inner product is the number a1b1 + a2b2 + p + anbn. For example taking the inner
product, we have
32
-1
0
5
6
44 ≥
¥ = 2 # 5 + 1- 12 # 6 + 0 # 1- 32 + 4 # 2 = 12
-3
2
SECTION 7.5
■
Matrix Operations: Getting Information from Data
613
Matrix Multiplication
If A is an m * n matrix and B is an n * k matrix, then their product AB is an
m * k matrix C.
AB = C
The entries of the matrix C are calculated as follows: The entry in the ith row
and jth column of the matrix C is the inner product of the ith row of A and the
jth column of B.
The following diagram shows that the entry cij in the ith row and jth column of
the product matrix AB is obtained by multiplying the entries in the ith row of A by
the corresponding entries in the jth column of B and then adding the results.
jth column of B
ith row of A
cٗ ٗ ٗd
ٗ
ٗ
ٗ
£
Entry in ith row and
jth column of AB
§ = c
cij d
The next example illustrates the process.
example 2
Multiplying Matrices
Calculate, if possible, the products AB and BA.
A= c
1
-1
3
d
0
B = c
-1
0
5
4
2
d
7
Solution
Inner numbers match,
so product is defined.
Since matrix A has dimension 2 * 2 and matrix B has dimension 2 * 3, the product
AB is defined and has dimension 2 * 3. We can write
AB = c
2* 2 2* 3
Outer numbers give dimension
of product: 2 * 3.
1
-1
3 -1
dc
0
0
2
d = c
7
5
4
?
?
?
?
?
?
d
where the question marks must be filled in by using the rule defining the product of
two matrices. If we define C = AB = 3cij 4 , then the entry c11 is the inner product of
the first row of A and the first column of B.
c
1
-1
3 -1
dc
0
0
5
4
2
d
7
1 # 1- 12 + 3 # 0 = - 1
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CHAPTER 7
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Systems of Equations and Data in Categories
Similarly, we calculate the remaining entries of the product as follows.
Entry
Inner product of
Value
Product matrix
c12
c
1
-1
3 -1
dc
0
0
5
4
2
d
7
1 # 5 + 3 # 4 = 17
c
-1
17
c13
c
1
-1
3 -1
dc
0
0
5
4
2
d
7
1 # 2 + 3 # 7 = 23
c
-1
17
23
d
c21
c
1
-1
3 -1
dc
0
0
5
4
2
d
7
1- 12 # 1- 12 + 0 # 0 = 1
c
-1
1
17
23
d
c22
c
1
-1
3 -1
dc
0
0
5
4
2
d
7
1- 12 # 5 + 0 # 4 = - 5
c
-1
1
17
-5
23
d
c23
c
1
-1
3 -1
dc
0
0
5
4
2
d
7
1- 12 # 2 + 0 # 7 = - 2
c
-1
1
17
-5
23
d
-2
d
Thus we have
Not equal, so product
not defined.
AB = c
-1
1
17
-5
23
d
-2
The product BA is not defined, however, because the dimensions of B and A are
2 *3
2*3 2*2
and
2 *2
The inner two numbers are not the same, so the rows and columns won’t match up
when we try to calculate the product.
■
■
NOW TRY EXERCISES 11 AND 13
Graphing calculators are capable of performing all the matrix operations. In the
next example we use a graphing calculator to multiply two matrices.
e x a m p l e 3 Multiplying Matrices (Graphing Calculator)
Calculate the products AB and BA.
3
A = £ 2
-1
0
2
1
5
-4§
2
1
B = £0
3
-1
7
8
5
6§
-2
Solution
Using a graphing calculator, we find the product AB.
Calculator output
[A] * [B]
[[18
37
5 ]
[-10 -20 30]
[5
24
-3]
Product AB
18
£ - 10
5
37
- 20
24
5
30 §
-3
SECTION 7.5
■
Matrix Operations: Getting Information from Data
615
Similarly, we find the product BA.
Calculator output
Product BA
[B] * [A]
[[-4 3
19 ]
[8
20 -16]
[27 14 -21]]
■
2
-4
£ 8
27
3
20
14
19
- 16 §
- 21
■
NOW TRY EXERCISE 19
■ Getting Information from Categorical Data
We will see in the next example how we can get information about categorical data
by multiplying matrices. (Compare to Example 5 in Section 7.4.)
example 4
Getting Information by Multiplying Matrices
Josh sells organic produce at an open-air market three days a week: Thursdays,
Fridays, and Saturdays. He sells oranges, broccoli, and beans. Matrix A tabulates the
number of pounds of produce that he obtains from his suppliers every week; he is always able to sell his complete supply by the end of each day. Matrix C gives the
prices per pound (in dollars) he charged on two different weeks. Find the product AC,
and interpret the entries of AC.
Produce
Oranges Broccoli
T
50
A = £ 70
45
T
20
35
15
Beans
T
10 d
30 § d
25 d
Thursday
Friday
Day
Saturday
Prices
Week 1 Week 2
T
0.90
C = £ 1.20
1.50
T
1.20 d
1.60 § d
1.90 d
Oranges
Broccoli Produce
Beans
Solution
We use a graphing calculator to find the product AC.
Calculator output
[A] * [C]
[[84
111
]
[150 197
]
[96
125.5]]
Product AC
84.00
£ 150.00
96.00
111.00
197.00 §
125.50
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CHAPTER 7
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Systems of Equations and Data in Categories
The product matrix gives us the revenue for each day of each week.
Week 1: A total of $84.00 of produce was sold on Thursday, $150.00 on
Friday, and $96.00 on Saturday.
Week 2: A total of $111.00 of produce was sold on Thursday, $197.00 on
Friday, and $125.50 on Saturday.
■
■
■
■
NOW TRY EXERCISE 27
7.5 Exercises
CONCEPTS
Fundamentals
1. We can add or (subtract) two matrices only if they have the same _______.
2. We can multiply two matrices only if the number of _______ in the first matrix is the
same as the number of _______ in the second matrix.
3. If A is a 3 * 3 matrix and B is a 2 * 3 matrix, which of the following matrix
multiplications are possible?
(a) AB
(b) BA
(c) AA
(d) BB
4. Fill in the missing entries in the product matrix.
3
£-1
1
1
2
3
-1
2
0§ £ 3
2
-2
3
-2
1
ٗ -7
-2
4
ٗ§
-1§ = £ 7 -7
ٗ -5 -5
0
Think About It
5. Which of the following operations can we perform on a matrix A of any dimension?
(a) A + A
(b) 2A
(c) AA
6. If A is a matrix for which AA is defined, what must be true about the dimension of A?
SKILLS
7–14
7. c
■
2
-5
1
9. 3 £ 4
1
2
11. £ 1
2
1
13. c
-1
Perform the matrix operation, or explain why the operation is not defined.
6
-1
d + c
3
6
-3
d
2
2
-1§
0
6
1
3§ £ 3
4 -2
2 1
dc
4 2
8. c
0
1
1
10. 2 £ 1
0
-2
6§
0
-2
2
12. c
3
d
-1
2
6
2
14. £ 0
1
1
2
d - c
0
1
1
1
1
0
1
1
3
0
1
1§ + £2
1
3
1
2
d£ 3
4
-2
-3
5
1§ c d
1
2
-1
d
-2
1
3
1
1§
1
-2
6§
0
SECTION 7.5
15–26
■
Matrix Operations: Getting Information from Data
617
Matrices A, B, C, D, E, F, G, and H are defined as follows. Perform the indicated
matrix operation(s), or explain why it cannot be performed.
A= c
2
0
D = 37
5
G= £ 6
-5
CONTEXTS
■
1
2
-5
d
7
B= c
34
1
E = £2§
0
-3
1
2
10
0§
2
H= c
3
1
-1
5
d
3
C = c
2
0
1
F = £0
0
- 52
2
0
1
0
0
d
-3
0
0§
1
1
d
-1
3
2
15. (a) B + C
(b) B + F
16. (a) C - B
(b) 2C - 6B
17. (a) 5A
(b) C - 5A
18. (a) 3B + 2C
(b) 2H + D
19. (a) AD
(b) DA
20. (a) DH
(b) HD
21. (a) AH
(b) HA
22. (a) BC
(b) BF
2
23. (a) GF
(b) GE
24. (a) B
25. (a) A2
(b) A3
26. (a) (DA)B
(b) F 2
(b) D(AB)
27. Fast-Food Sales A small fast-food chain with restaurants in Santa Monica, Long
Beach, and Anaheim sells only hamburgers, hot dogs, and milk shakes. On a certain
day, sales were distributed according to matrix A. The price of each item (in dollars) is
given by matrix C.
(a) Calculate the product matrix CA.
(b) What is the total profit from the Long Beach restaurant?
(c) What is the total profit (from all three restaurants)?
Number of items sold
Santa
Long
Monica
Beach
Anaheim
T
T
4000
A = £ 400
700
T
1000
300
500
3500 d Hamburgers
200 § d Hot dogs
Item
9000 d Milk shakes
Item
Hamburger Hot dog
T
C = 3 0.90
T
0.80
Milk shake
T
1.104 d Price
28. Car-Manufacturing Profits A specialty car manufacturer has plants in Auburn,
Biloxi, and Chattanooga. Three models are produced, with daily production given in
matrix A. Because of a wage increase, February profits are lower than January profits.
The profit (in dollars) per car is tabulated by model in matrix B.
(a) Calculate the product AB.
(b) Assuming that all cars produced were sold, what was the daily profit in January
from the Biloxi plant?
(c) What was the total daily profit (from all three plants) in February?
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CHAPTER 7
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Systems of Equations and Data in Categories
Cars produced each day
Model K
Model R
Model W
T
T
12
A= £ 4
8
10
4
9
T
0 d Auburn
City
20 § d Biloxi
12 d Chattanooga
January February
1000
B = £ 2000
1500
500 d Model K
1200 § d Model R
1000 d Model W
29. Canning Tomato Products Jaeger Foods produces tomato sauce and tomato paste,
canned in small, medium, large, and giant sized cans. The matrix A gives the size (in
ounces) of each container. The matrix B tabulates one day’s production of tomato sauce
and tomato paste.
(a) Calculate the product AB.
(b) Interpret the entries in the product matrix AB.
Size
Small Medium Large
T
A = 36
Giant
T
T
T
10
14
284
d Ounces
Cans of sauce Cans of paste
T
T
2000
3000
B= ≥
2500
1000
2500
1500
¥
1000
500
d
d
d
d
Small
Medium
Large
Giant
30. Produce Sales A farmer’s three children, Amy, Beth, and Chad, run three roadside
produce stands during the summer months. One weekend they all sell watermelons,
yellow squash, and tomatoes. Matrices A and B tabulate the number of pounds of each
product sold by each sibling on Saturday and Sunday. Matrix C gives the price per
pound (in dollars) for each type of produce that they sell. Perform each of the following
matrix operations, and interpret the entries in each result.
(a) AC
(b) BC
(c) A + B
(d) 1A + B2 C
Melons
Saturday
Squash
T
T
120
A = £ 40
60
50
25
30
Melons
Sunday
Squash
Tomatoes
T
T
T
60
20
25
30 d Amy
20 § d Beth
30 d Chad
100
B = £ 35
60
Tomatoes
T
60 d Amy
30 § d Beth
20 d Chad
Price per pound
0.10 d Melons
C = £ 0.50 § d Squash
1.00 d Tomatoes
SECTION 7.6
2
Matrix Equations: Solving a Linear System
■
619
7.6 Matrix Equations: Solving a Linear System
■
The Inverse of a Matrix
■
Matrix Equations
■
Modeling with Matrix Equations
IN THIS SECTION… we express a system of equations as a single matrix equation and
then solve the system by solving the matrix equation.
In the preceding section we saw that when the dimensions are appropriate, matrices
can be added, subtracted, and multiplied. In this section we investigate “division” of
matrices. With this operation we can solve equations that involve matrices.
2
■ The Inverse of a Matrix
First, we define identity matrices, which play the same role for matrix multiplication that the number 1 does for ordinary multiplication of numbers; that is,
1 # a = a # 1 = a for all numbers a. In the following definition the term main diagonal refers to the entries of a square matrix whose row and column numbers
are the same. These entries stretch diagonally down the matrix, from top left to
bottom right.
Identity Matrix
The identity matrix In is the n * n matrix for which each main diagonal
entry is a 1 and for which all other entries are 0.
Thus, the 2 * 2, 3 * 3, and 4 * 4 identity matrices are
I2 = c
1
0
0
d
1
1
0
I4 =
0
0
0
1
0
0
1
I3 = Ê 0
0
0
1
0
0
0Đ
1
0
0
1
0
0
0
Ơ
0
1
Identity matrices behave like the number 1 in the sense that
A # In = A
and
B # In = B
whenever these products are defined.
The following matrix products show how multiplying a matrix by an identity
matrix of the appropriate dimension leaves the matrix unchanged.
c
1
0
-1
£ 12
-2
0
3
dc
1 -1
7
1
0
1
2
1
3§ £0
7 0
6
3
d = c
7
-1
5
2
0
1
0
0
-1
0 § = £ 12
1
-2
6
d
7
5
2
7
1
0
1
2
3§
7
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CHAPTER 7
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Systems of Equations and Data in Categories
If A and B are n * n matrices, and if AB = BA = In, then we say that B is the
inverse of A, and we write B = A -1. The concept of the inverse of a matrix is analogous to that of the reciprocal of a real number.
Inverse of a Matrix
Let A be a square n * n matrix. If there exists an n * n matrix A -1 with the
property that
AA -1 = A -1A = In
then we say that A -1 is the inverse of A.
e x a m p l e 1 Verifying That a Matrix Is an Inverse
Verify that matrix B is the inverse of matrix A.
A = c
2
5
3
d
1
B = c
and
3 -1
d
-5
2
Solution
We perform the matrix multiplications to show that AB = I and BA = I.
c
2
5
c
3
-5
1
3
dc
3 -5
-1 2
dc
2 5
-1
2 # 3 + 11- 52
d = c #
2
5 3 + 31- 52
21- 12 + 1 # 2
1
d = c
#
51- 12 + 3 2
0
0
d
1
1
3 # 2 + 1- 125
d = c
3
1- 522 + 2 # 5
3 # 1 + 1- 123
1
d = c
#
1- 521 + 2 3
0
0
d
1
■
NOW TRY EXERCISE 3
■
The inverse of a matrix can be found by using a process that involves elementary row operations. This process is programmed into graphing calculators. On
TI-83 calculators, matrices are stored in memory by using names such as [A], [B],
[C], and so on. To find the inverse of [A], we key in
[A]
XϪ1
ENTER
e x a m p l e 2 Finding the Inverse of a Matrix
Find the inverse of matrix A.
1
A= £ 2
-3
-2
-3
6
-4
-6§
15
Solution
A graphing calculator gives the following output for the inverse of matrix A.