Holt McDougal Algebra 2 4-1 Matrices and Data 4-1 Matrices and Data Holt Algebra 2 Warm Up Warm Up...
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Transcript of Holt McDougal Algebra 2 4-1 Matrices and Data 4-1 Matrices and Data Holt Algebra 2 Warm Up Warm Up...
Holt McDougal Algebra 2
4-1 Matrices and Data4-1 Matrices and Data
Holt Algebra 2
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
4-1 Matrices and Data
Warm UpDistribute.
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
Holt McDougal Algebra 2
4-1 Matrices and Data
Use matrices to display mathematical and real-world data.
Find sums, differences, and scalar products of matrices.
Objectives
Holt McDougal Algebra 2
4-1 Matrices and Data
matrixdimensionsentryaddressscalar
Vocabulary
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
The prices for different sandwiches are presented at right.
Example 1: Displaying Data in Matrix Form
6 in 9 in
Roast beef $3.95 $5.95
Turkey $3.75 $5.60
Tuna $3.50 $5.25
A. Display the data in matrix form. P =
3.95 5.95
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
C. What is entry P32? What does is represent?
D. What is the address of the entry 5.95?
The entry at P32, in row 3 column 2, is 5.25. It is the price of a 9 in. tuna sandwich.
The entry 5.95 is at P12.
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
Holt McDougal Algebra 2
4-1 Matrices and Data
Use matrix M to answer the questions below.
Check It Out! Example 1
a. What are the dimensions of M? 3 4
11
m14 and m23
b. What is the entry at m32?
c. The entry 0 appears at what two addresses?
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
Add or subtract, if possible.
Example 2A: Finding Matrix Sums and Differences
W + Y
Add each corresponding entry.
W = , 3 –2
1 0Y = ,
1 4
–2 3X = ,
4 7 2
5 1 –1Z =
2 –2 3
1 0 4
W + Y =
3 –2
1 0+
1 4
–2 3=
3 + 1 –2 + 4
1 + (–2) 0 + 3
4 2
–1 3=
Holt McDougal Algebra 2
4-1 Matrices and Data
X – Z
Subtract each corresponding entry.
Example 2B: Finding Matrix Sums and Differences
W = , 3 –2
1 0Y = ,
1 4
–2 3X = ,
4 7 2
5 1 –1Z =
2 –2 3
1 0 4
Add or subtract, if possible.
X – Z =4 7 2
5 1 –1
2 –2 3
1 0 4–
2 9 –1
4 1 –5 =
Holt McDougal Algebra 2
4-1 Matrices and Data
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.
Example 2C: Finding Matrix Sums and Differences
W = , 3 –2
1 0Y = ,
1 4
–2 3X = ,
4 7 2
5 1 –1Z =
2 –2 3
1 0 4
Add or subtract, if possible.
Holt McDougal Algebra 2
4-1 Matrices and Data
Add or subtract if possible.
Check It Out! Example 2A
B + D
A = ,
4 –2
–3 10
2 6
B = , 4 –1 –5
3 2 8C = ,
3 2
0 –9
–5 14
D = 0 1 –3
3 0 10
Add each corresponding entry.
B + D =
+ 4 –1 –5
3 2 8
0 1 –3
3 0 10
4 + 0 –1 + 1 –5 + (–3)
3 + 3 2 + 0 8 + 10=
4 0 –8
6 2 18
Holt McDougal Algebra 2
4-1 Matrices and Data
B – A
Add or subtract if possible.
Check It Out! Example 2B
A = ,
4 –2
–3 10
2 6
B = , 4 –1 –5
3 2 8C = ,
3 2
0 –9
–5 14
D = 0 1 –3
3 0 10
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
4-1 Matrices and Data
D – B Subtract corresponding entries.
Add or subtract if possible.
Check It Out! Example 2C
A = ,
4 –2
–3 10
2 6
B = , 4 –1 –5
3 2 8C = ,
3 2
0 –9
–5 14
D = 0 1 –3
3 0 10
0 1 –3
3 0 10
4 –1 –5
3 2 8–
–4 2 2
0 –2 2= D – B =
Holt McDougal Algebra 2
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
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
Use a scalar product to find the prices if a 10% discount is applied to the prices above.
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 15
8 17.5
9 20
10 22.5
– 0.1 =
7.5 15
8 17.5
9 20
10 22.5
0.75 1.5
0.8 1.75
0.9 2
1 2.25
–
6.75 13.50
7.20 15.75
8.10 18.00
9.00 20.25
Holt McDougal Algebra 2
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
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.
150 87.5
125 70
200 112.5
– 0.2 =150 87.5
125 70
200 112.5
150 87.5
125 70
200 112.5
30 17.5
25 14
40 22.5
–
120 70
100 56
160 90
Holt McDougal Algebra 2
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
Evaluate 3P — Q, if possible.
P =
3 –2
1 0
2 –1
Q=4 7 2
5 1 –1R =
1 4
–2 3
0 4
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
Evaluate 3R — P, if possible.
P =
3 –2
1 0
2 –1
Q=4 7 2
5 1 –1R =
1 4
–2 3
0 4
= 3
1 4
–2 3
0 4
–
3 –2
1 0
2 –1
=
3(1) 3(4)
3(–2) 3(3)
3(0) 3(4)
–
3 –2
1 0
2 –1
=
3 12
–6 9
0 12
–
3 –2
1 0
2 –1
0 14
–7 9
–2 13
Holt McDougal Algebra 2
4-1 Matrices and Data
Check It Out! Example 4a
Evaluate 3B + 2C, if possible.
D = [6 –3 8]A =
4 –2
–3 10C =
3 2
0 –9 B =
4 –1 –5
3 2 8
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
D = [6 –3 8]A =
4 –2
–3 10C =
3 2
0 –9 B =
4 –1 –5
3 2 8
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
0 27 =
–1 –10
–6 47
Holt McDougal Algebra 2
4-1 Matrices and Data
= [6 –3 8] + 0.5[6 –3 8]
Check It Out! Example 4c
D = [6 –3 8]A =
4 –2
–3 10C =
3 2
0 –9 B =
4 –1 –5
3 2 8
Evaluate D + 0.5D, if possible.
= [6 –3 8] + [0.5(6) 0.5(–3) 0.5(8)]
= [6 –3 8] + [3 –1.5 4]
= [9 –4.5 12]
Holt McDougal Algebra 2
4-1 Matrices and Data
Holt McDougal Algebra 2
4-1 Matrices and Data
Lesson Quiz
1. What are the dimensions of A?
2. What is entry D12?
Evaluate if possible.
3. 2A — C 4. C + 2D 5. 10(2B + D)
–2
3 2
Not possible