1. 4-1 Matrices and Data Warm Up Lesson Presentation Lesson Quiz
Holt McDougal Algebra 2
Holt Algebra 2

2. Warm Up
Distribute.
1. 3(2x + y + 3z)
2. –1(x – y + 2)
State the property illustrated.
3. (a + b) + c = a + (b + c)
4. p + q = q + p
6x + 3y + 9z
–x + y – 2
Associative Property of Addition
Commutative Property of Addition

3. Objectives Use matrices to display mathematical and real-world data.
Find sums, differences, and scalar products of matrices.

4. Vocabulary
matrix
dimensions
entry
address
scalar

5. 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.

6. Matrix A has two rows and three columns
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.

7. 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 is located in row 2 column 1, so a21 is

8. Example 1: Displaying Data in Matrix Form
Roast beef
$3.95
$5.95
Turkey
$3.75
$5.60
Tuna
$3.50
$5.25
The prices for different sandwiches are presented at right.
P =
A. Display the data in matrix form.
B. What are the dimensions of P?
P has three rows and two columns, so it is a 3  2 matrix.

9. Example 1: Displaying Data in Matrix Form
Roast beef
$3.95
$5.95
Turkey
$3.75
$5.60
Tuna
$3.50
$5.25
The prices for different sandwiches are presented at right.
C. What is entry P32? What does is represent?
The entry at P32, in row 3 column 2, is 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.

10. Use matrix M to answer the questions below.
Check It Out! Example 1
Use matrix M to answer the questions below.
a. What are the dimensions of M?
3  4
b. What is the entry at m32? 11
c. The entry 0 appears at what two addresses?
m14 and m23

11. You can add or subtract two matrices only if they have the same dimensions.

12. Example 2A: Finding Matrix Sums and Differences
Add or subtract, if possible.
3 –2
1 0 –1
1 4
–2 3
2 –2 3
W = ,
X = ,
Y = ,
Z =
W + Y
Add each corresponding entry.
3 –2
1 0 +
1 4
–2 3 =
–2 + 4
1 + (–2)
4 2
–1 3 =
W + Y =

13. Example 2B: Finding Matrix Sums and Differences
Add or subtract, if possible.
3 –2
1 0 –1
1 4
–2 3
2 –2 3
W = ,
X = ,
Y = ,
Z =
X – Z
Subtract each corresponding entry. –1
2 –2 3 – –1 –5 =
X – Z =

14. Example 2C: Finding Matrix Sums and Differences
Add or subtract, if possible.
3 –2
1 0 –1
1 4
–2 3
2 –2 3
W = ,
X = ,
Y = ,
Z =
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.

15. Check It Out! Example 2A
Add or subtract if possible.
4 –2
–3 10
2 6
0 –9
–5 14
4 –1 –5 –3
A = ,
B = ,
C = ,
D =
B + D
Add each corresponding entry.
B + D = –8 +
4 –1 –5 –3
– –5 + (–3) =

16. Check It Out! Example 2B
Add or subtract if possible.
4 –2
–3 10
2 6
0 –9
–5 14
4 –1 –5 –3
A = ,
B = ,
C = ,
D =
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.

17. Check It Out! Example 2C
Add or subtract if possible.
4 –2
–3 10
2 6
0 –9
–5 14
4 –1 –5 –3
A = ,
B = ,
C = ,
D =
D – B
Subtract corresponding entries. –3
4 –1 –5 – –
0 –2 2 =
D – B =

18. You can multiply a matrix by a number, called a scalar
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.

19. 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
X-Large
$10.00
$22.50
You can multiply by 0.1 and subtract from the original numbers. –
– 0.1 =

20. 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

21. You can multiply by 0.2 and subtract from the original numbers.
Check It Out! Example 3
Ticket Service Prices
Days
Plaza
Balcony
1—2
$150
$87.50
3—8
$125
$70.00
9—10
$200
$112.50
Use a scalar product to find the prices if a 20% discount is applied to the ticket service prices.
You can multiply by 0.2 and subtract from the original numbers. –
– 0.2 =

22. Discount Ticket Service Prices
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

23. Example 4A: Simplifying Matrix Expressions
3 –2
1 0
2 –1
1 4
–2 3
0 4 –1
P = Q=
R =
Evaluate 3P — Q, if possible.
P and Q do not have the same dimensions; they cannot be subtracted after the scalar products are found.

24. Example 4B: Simplifying Matrix Expressions
3 –2
1 0
2 –1
1 4
–2 3
0 4 –1
P = Q=
R =
Evaluate 3R — P, if possible.
= 3
1 4
–2 3
0 4 –
3 –2
1 0
2 –1 =
–6 9 –
3 –2
1 0
2 –1
3(1) 3(4)
3(–2) 3(3)
3(0) 3(4) –
3 –2
1 0
2 –1
–7 9
–2 13 =

25. Check It Out! Example 4a
4 –2
–3 10
4 –1 –5
3 2
0 –9
D = [6 –3 8]
A =
B =
C =
Evaluate 3B + 2C, if possible.
B and C do not have the same dimensions; they cannot be added after the scalar products are found.

26. Check It Out! Example 4b
4 –2
–3 10
4 –1 –5
3 2
0 –9
D = [6 –3 8]
A =
B =
C =
Evaluate 2A – 3C, if possible.
4 –2
–3 10
= 2
– 3
3 2
0 –9
2(4) 2(–2)
2(–3) 2(10) = +
–3(3) –3(2)
–3(0) –3(–9)
8 –4
–6 20 = +
–9 –6 =
–1 –10
–6 47

27. Check It Out! Example 4c
4 –2
–3 10
4 –1 –5
3 2
0 –9
D = [6 –3 8]
A =
B =
C =
Evaluate D + 0.5D, if possible.
= [6 –3 8] + 0.5[6 –3 8]
= [6 –3 8] + [0.5(6) 0.5(–3) 0.5(8)]
= [6 –3 8] + [3 – ]
= [9 – ]

29. Lesson Quiz
1. What are the dimensions of A?
2. What is entry D12?
Evaluate if possible.
3. 2A — C 4. C + 2D (2B + D)
3  2 –2
Not possible