Download A = , 4 –2

4-1
4-1 Matrices
Matricesand
andData
Data
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
4-1 Matrices and Data
Warm Up
Distribute.
1. 3(2x + y + 3z) 6x + 3y + 9z
2. –1(x – y + 2) –x + y – 2
State the property illustrated.
3. (a + b) + c = a + (b + c)
Associative Property of Addition
4. p + q = q + p
Commutative Property of Addition
Holt McDougal Algebra 2
4-1 Matrices and Data
Objectives
Use matrices to display mathematical
and real-world data.
Find sums, differences, and scalar
products of matrices.
Holt McDougal Algebra 2
4-1 Matrices and Data
Vocabulary
matrix
dimensions
entry
address
scalar
Holt McDougal Algebra 2
4-1 Matrices and Data
The table shows the top scores for girls in barrel
racing at the 2004 National High School Rodeo
finals. The data can be presented in a table or a
spreadsheet as rows and columns of numbers. You
can also use a matrix to show table data. A matrix
is a rectangular array of numbers enclosed in
brackets.
Holt McDougal Algebra 2
4-1 Matrices and Data
Matrix A has two rows and three columns. A matrix
with m rows and n columns has dimensions m  n,
read “m by n,” and is called an m  n matrix. A has
dimensions 2  3. Each value in a matrix is called an
entry of the matrix.
Holt McDougal Algebra 2
4-1 Matrices and Data
The address of an entry is its location in a matrix,
expressed by using the lower case matrix letter with
row and column number as subscripts. The score
16.206 is located in row 2 column 1, so a21 is 16.206.
Holt McDougal Algebra 2
4-1 Matrices and Data
Example 1: Displaying Data in Matrix Form
The prices for different
sandwiches are
presented at right.
A. Display the data in
matrix form.
6 in
9 in
Roast beef
$3.95
$5.95
Turkey
$3.75
$5.60
Tuna
$3.50
$5.25
3.95
5.95
P = 3.75
5.60
3.50
5.25
B. What are the dimensions of P?
P has three rows and two columns, so it is a 3  2
matrix.
Holt McDougal Algebra 2
4-1 Matrices and Data
Example 1: Displaying Data in Matrix Form
The prices for different
sandwiches are
presented at right.
6 in
9 in
Roast beef
$3.95
$5.95
Turkey
$3.75
$5.60
Tuna
$3.50
$5.25
C. What is entry P32? What does is represent?
The entry at P32, in row 3 column 2, is 5.25. It is
the price of a 9 in. tuna sandwich.
D. What is the address of the entry 5.95?
The entry 5.95 is at P12.
Holt McDougal Algebra 2
4-1 Matrices and Data
Check It Out! Example 1
Use matrix M to
answer the
questions below.
a. What are the dimensions of M?
b. What is the entry at m32?
34
11
c. The entry 0 appears at what two addresses?
m14 and m23
Holt McDougal Algebra 2
4-1 Matrices and Data
You can add or subtract two matrices only
if they have the same dimensions.
Holt McDougal Algebra 2
4-1 Matrices and Data
Example 2A: Finding Matrix Sums and Differences
Add or subtract, if possible.
W=
3 –2
1
0
,X=
4
7
2
5
1 –1
, Y=
1
4
–2 3
, Z=
2 –2 3
1
0
4
W+Y
Add each corresponding entry.
W+Y=
3 –2
1
0
Holt McDougal Algebra 2
+
1
4
–2 3
=
3+1
1 + (–2)
–2 + 4
0+3
=
4
2
–1 3
4-1 Matrices and Data
Example 2B: Finding Matrix Sums and Differences
Add or subtract, if possible.
W=
3 –2
1
0
,X=
4
7
2
5
1 –1
, Y=
1
4
–2 3
, Z=
X–Z
Subtract each corresponding entry.
X–Z=
4
7
5
1 –1
Holt McDougal Algebra 2
2
–
2 –2
3
1
4
0
=
2
9
–1
4
1
–5
2 –2 3
1
0
4
4-1 Matrices and Data
Example 2C: Finding Matrix Sums and Differences
Add or subtract, if possible.
W=
3 –2
1
0
,X=
4
7
2
5
1 –1
, Y=
1
4
–2 3
, Z=
2 –2 3
1
0
4
X+Y
X is a 2  3 matrix, and Y is a 2  2 matrix.
Because X and Y do not have the same dimensions,
they cannot be added.
Holt McDougal Algebra 2
4-1 Matrices and Data
Check It Out! Example 2A
Add or subtract if possible.
4
–2
A = –3 10 , B =
2
B+D
6
4
–1 –5
3
2
8
3
,C = 0
2
–9 , D =
–5 14
0
1 –3
3
0
Add each corresponding entry.
B+D=
4
–1 –5
3
2
8
+
0
1 –3
3
0
Holt McDougal Algebra 2
10
=
0 –8
44+ 0
–1 + 1 –5 + (–3)
2
36+ 3
218
+0
8 + 10
10
4-1 Matrices and Data
Check It Out! Example 2B
Add or subtract if possible.
4
–2
A = –3 10 , B =
2
6
4
–1 –5
3
2
8
3
,C = 0
2
–9 , D =
–5 14
B–A
B is a 2  3 matrix, and A is a 3  2 matrix.
Because B and A do not have the same
dimensions, they cannot be subtracted.
Holt McDougal Algebra 2
0
1 –3
3
0
10
4-1 Matrices and Data
Check It Out! Example 2C
Add or subtract if possible.
4
–2
A = –3 10 , B =
2
6
D–B
D–B=
4
–1 –5
3
2
8
3
,C = 0
2
–9 , D =
–5 14
Subtract corresponding entries.
0
1
–3
3
0
10
Holt McDougal Algebra 2
–
4
–1 –5
3
2
8
=
–4 2
0
2
–2 2
0
1 –3
3
0
10
4-1 Matrices and Data
You can multiply a matrix by a number, called a scalar.
To find the product of a scalar and a matrix, or the
scalar product, multiply each entry by the scalar.
Holt McDougal Algebra 2
4-1 Matrices and Data
Example 3: Business Application
Use a scalar
product to find
the prices if a
10% discount is
applied to the
prices above.
Shirt Prices
T-shirt
Sweatshirt
Small
$7.50
$15.00
Medium
$8.00
$17.50
Large
$9.00
$20.00
$10.00
$22.50
X-Large
You can multiply by 0.1 and subtract from the
original numbers.
7.5
15
8
17.5
9
20
10
22.5
7.5
– 0.1 8
Holt McDougal Algebra 2
15
17.5
9
20
10
22.5
=
6.75
13.50
7.5 15
7.20
15.75
8
17.5
–
8.10
18.00
9
20
9.00
20.25
10 22.5
0.75
1.5
0.8
1.75
0.9
2
1
2.25
4-1 Matrices and Data
Example 3 Continued
The discount prices are shown in the table.
Discount Shirt Prices
T-shirt Sweatshirt
Small
$6.75
$13.50
Medium
$7.20
$15.75
Large
$8.10
$18.00
X-large
$9.00
$20.25
Holt McDougal Algebra 2
4-1 Matrices and Data
Check It Out! Example 3
Use a scalar product
to find the prices if a
20% discount is
applied to the ticket
service prices.
Ticket Service Prices
Days
Plaza
Balcony
1—2
$150
$87.50
3—8
$125
$70.00
9—10
$200
$112.50
You can multiply by 0.2 and subtract from the
original numbers.
150
87.5
125
70
200
112.5
150
87.5
– 0.2 125 70
Holt McDougal Algebra 2
200
112.5
150
120 87.5
70
= 125
100 70
56
200
160 112.5
90
30
17.5
– 25 14
40
22.5
4-1 Matrices and Data
Check It Out! Example 3 Continued
Discount Ticket Service Prices
Days
Plaza
Balcony
1—2
$120
$70
3—8
$100
$56
9—10
$160
$90
Holt McDougal Algebra 2
4-1 Matrices and Data
Example 4A: Simplifying Matrix Expressions
3
P= 1
2
–2
0
–1
Q=
4
7
2
5
1 –1
1
4
R = –2 3
0
4
Evaluate 3P — Q, if possible.
P and Q do not have the same dimensions; they
cannot be subtracted after the scalar products are
found.
Holt McDougal Algebra 2
4-1 Matrices and Data
Example 4B: Simplifying Matrix Expressions
3
P= 1
2
–2
Q=
0
–1
4
7
2
5
1 –1
1
4
R = –2 3
0
4
Evaluate 3R — P, if possible.
1
4
= 3 –2 3
0
3 –2
–
4
Holt McDougal Algebra 2
1
0
2 –1
3(1)
3
0 12
14
3(4) 3 –2
3 –2
= –6
– –1
3(–2)
–79 9
3(3)
0
1
0
3(0)
0
–21213
3(4) 2 –1
2 –1
4-1 Matrices and Data
Check It Out! Example 4a
A=
4
–2
–3 10
B=
4 –1 –5
3
2
8
C=
3
2
0 –9
D = [6 –3 8]
Evaluate 3B + 2C, if possible.
B and C do not have the same dimensions; they
cannot be added after the scalar products are found.
Holt McDougal Algebra 2
4-1 Matrices and Data
Check It Out! Example 4b
A=
4
–2
–3 10
B=
4 –1 –5
3
2
C=
8
3
2
0 –9
D = [6 –3 8]
Evaluate 2A – 3C, if possible.
=2
4
–2
–3 10
–3
3
2
0 –9
=
=
Holt McDougal Algebra 2
2(4)
2(–2)
2(–3) 2(10)
8
–4
–6 20
+
+
–3(3) –3(2)
–3(0) –3(–9)
–9 –6
0
27
=
–1 –10
–6 47
4-1 Matrices and Data
Check It Out! Example 4c
A=
4
–2
–3 10
B=
4 –1 –5
3
2
8
C=
3
2
0 –9
Evaluate D + 0.5D, if possible.
= [6 –3
8] + 0.5[6 –3
= [6 –3
8] + [0.5(6) 0.5(–3)
= [6 –3
8] + [3 –1.5
= [9 –4.5
12]
Holt McDougal Algebra 2
8]
4]
0.5(8)]
D = [6 –3 8]
4-1 Matrices and Data
Holt McDougal Algebra 2
4-1 Matrices and Data
Lesson Quiz
1. What are the dimensions of A? 3  2
2. What is entry D12? –2
Evaluate if possible.
3. 2A — C
4. C + 2D
Not possible
Holt McDougal Algebra 2
5. 10(2B + D)