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

Use matrices to display mathematical and real-world data.

Find sums, differences, and scalar products of matrices.

Objectives

matrixdimensionsentryaddressscalar

Vocabulary

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.

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.

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.

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.

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

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?

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

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=

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 =

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.

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

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.

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 =

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.

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

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

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

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

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.

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

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.

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

= [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]

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