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