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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

Use matrices to display mathematical and real-world data.

Find sums, differences, and scalar products of matrices.

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 dimensionsm n, read “m by n,” and is called an mn matrix. A has dimensions 2 3. Each value in a matrix is called an entry of the matrix.

The rows and n columns has 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.

3.95 5.95 rows and n columns has

3.75 5.60

3.50 5.25

P =

Example 1: Displaying Data in Matrix Form

The prices for different sandwiches are presented at right.

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.

Example 1: Displaying Data in Matrix Form rows and n columns has

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 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.

Check It Out! rows and n columns has 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

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

3 –2 same dimensions.

1 0

3 + 1–2 + 4

1 + (–2)0 + 3

4 2

–1 3

+

=

=

1 4

–2 3

Example 2A: Finding Matrix Sums and Differences

Add or subtract, if possible.

3 –2

1 0

4 7 2

5 1 –1

1 4

–2 3

2 –2 3

1 0 4

W = ,

X = ,

Y = ,

Z =

W + Y

Add each corresponding entry.

W + Y =

2 –2 3 same dimensions.

1 0 4

4 7 2

5 1 –1

2 9 –1

4 1 –5

–

=

Example 2B: Finding Matrix Sums and Differences

Add or subtract, if possible.

3 –2

1 0

4 7 2

5 1 –1

1 4

–2 3

2 –2 3

1 0 4

W = ,

X = ,

Y = ,

Z =

X – Z

Subtract each corresponding entry.

X – Z =

Example 2C: Finding Matrix Sums and Differences same dimensions.

Add or subtract, if possible.

3 –2

1 0

4 7 2

5 1 –1

1 4

–2 3

2 –2 3

1 0 4

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.

4 0 –8 same dimensions.

6 2 18

4 –1 –5

3 2 8

0 1 –3

3 0 10

4 + 0 –1 + 1 –5 + (–3)

3 + 3 2 + 0 8 + 10

+

=

Check It Out! Example 2A

Add or subtract if possible.

4 –2

–3 10

2 6

3 2

0 –9

–5 14

4 –1 –5

3 2 8

0 1 –3

3 0 10

A = ,

C = ,

D =

B = ,

B + D

Add each corresponding entry.

B + D =

Check It Out! same dimensions. Example 2B

Add or subtract if possible.

4 –2

–3 10

2 6

3 2

0 –9

–5 14

4 –1 –5

3 2 8

0 1 –3

3 0 10

A = ,

C = ,

D =

B = ,

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.

0 1 –3 same dimensions.

3 0 10

4 –1 –5

3 2 8

–4 2 2

0 –2 2

–

=

Check It Out! Example 2C

Add or subtract if possible.

4 –2

–3 10

2 6

3 2

0 –9

–5 14

4 –1 –5

3 2 8

0 1 –3

3 0 10

A = ,

C = ,

D =

B = ,

D – B

Subtract corresponding entries.

D – B =

You can multiply a matrix by a number, called a same dimensions.scalar. To find the product of a scalar and a matrix, or the scalar product, multiply each entry by the scalar.

6.75 13.50 same dimensions.

7.20 15.75

8.10 18.00

9.00 20.25

7.5 15

8 17.5

9 20

10 22.5

7.5 15

8 17.5

9 20

10 22.5

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

–

Example 3: Business Application

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.

– 0.1

=

Example 3 Continued same dimensions.

The discount prices are shown in the table.

30 17.5 same dimensions.

25 14

40 22.5

150 87.5

125 70

200 112.5

150 87.5

125 70

200 112.5

150 87.5

125 70

200 112.5

–

120 70

100 56

160 90

Check It Out! Example 3

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

=

Check It Out! same dimensions. Example 3 Continued

Example 4A: Simplifying Matrix Expressions same dimensions.

3 –2

1 0

2 –1

1 4

–2 3

0 4

4 7 2

5 1 –1

P =

R =

Q=

Evaluate 3P — Q, if possible.

P and Q do not have the same dimensions; they cannot be subtracted after the scalar products are found.

3 12 same dimensions.

–6 9

0 12

1 4

–2 3

0 4

3 –2

1 0

2 –1

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

–

–

–

0 14

–7 9

–2 13

Example 4B: Simplifying Matrix Expressions

3 –2

1 0

2 –1

1 4

–2 3

0 4

4 7 2

5 1 –1

P =

R =

Q=

Evaluate 3R — P, if possible.

=

Check It Out! same dimensions. Example 4a

4 –2

–3 10

4 –1 –5

3 2 8

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.

4 –2 same dimensions.

–3 10

3 2

0 –9

= 2

–3

8 –4

–6 20

–9 –6

0 27

–1 –10

–6 47

2(4) 2(–2)

2(–3) 2(10)

–3(3) –3(2)

–3(0) –3(–9)

=

+

=

=

+

Check It Out! Example 4b

4 –2

–3 10

4 –1 –5

3 2 8

3 2

0 –9

D = [6 –3 8]

A =

B =

C =

Evaluate 2A – 3C, if possible.

Check It Out! same dimensions. Example 4c

4 –2

–3 10

4 –1 –5

3 2 8

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 –1.5 4]

= [9 –4.5 12]

Lesson Quiz same dimensions.

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)

3 2

–2

Not possible

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