Holt Algebra 2

4-1 Matrices and Data 4-1 Matrices and Data

Holt Algebra 2

Warm Up

Lesson Presentation

Lesson Quiz

Holt 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.

Holt Algebra 2

4-1 Matrices and Data

You can also use a matrix to show table data. A matrix is a rectangular array of numbers enclosed in brackets.

Holt Algebra 2

4-1 Matrices and Data

Use matrices to display mathematical and real-world data.

Find sums, differences, scalar products and products of matrices.

Objectives

Holt Algebra 2

4-1 Matrices and Data

Holt Algebra 2

4-1 Matrices and Data

Holt Algebra 2

4-1 Matrices and Data

Does the dimension matter?

Holt Algebra 2

4-1 Matrices and Data

matrix

dimensions

entry

address

scalar

Vocabulary

Holt Algebra 2

4-1 Matrices and Data

matrix

dimensions

entry

address

scalar

Vocabulary

Holt 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 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 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 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 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 Algebra 2

4-1 Matrices and Data

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

Holt Algebra 2

4-1 Matrices and Data

Add or subtract, if possible.

Example 2A: Finding Matrix Sums and Differences

W + Y

W = , 3 –2

1 0 Y = ,

1 4

–2 3 X = ,

4 7 2

5 1 –1 Z =

2 –2 3

1 0 4

Holt Algebra 2

4-1 Matrices and Data

X – Z

Example 2B: Finding Matrix Sums and Differences

W = , 3 –2

1 0 Y = ,

1 4

–2 3 X = ,

4 7 2

5 1 –1 Z =

2 –2 3

1 0 4

Add or subtract, if possible.

Holt Algebra 2

4-1 Matrices and Data

X + Y

Example 2C: Finding Matrix Sums and Differences

W = , 3 –2

1 0 Y = ,

1 4

–2 3 X = ,

4 7 2

5 1 –1 Z =

2 –2 3

1 0 4

Add or subtract, if possible.

Holt 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 8 C = ,

3 2

0 –9

–5 14

D = 0 1 –3

3 0 10

4 + 0 –1 + 1 –5 + (–3)

3 + 3 2 + 0 8 + 10

Holt 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 8 C = ,

3 2

0 –9

–5 14

D = 0 1 –3

3 0 10

Holt Algebra 2

4-1 Matrices and Data

D – B

Add or subtract if possible.

Check It Out! Example 2C

A = ,

4 –2

–3 10

2 6

B = , 4 –1 –5

3 2 8 C = ,

3 2

0 –9

–5 14

D = 0 1 –3

3 0 10

Holt Algebra 2

4-1 Matrices and Data

Write a short summary to explain how to add matrices.

Holt Algebra 2

4-1 Matrices and Data

Write a short summary to explain how to subtract matrices.

Holt Algebra 2

4-1 Matrices and Data

4 –2

–3 10

2 6

A =

Scalar Multiplication

3A = ½ A =

Holt Algebra 2

4-1 Matrices and Data

Write a short summary to explain how to Multiply a matrix by a scalar (constant) value.

Holt Algebra 2

4-1 Matrices and Data

2x-5 4 = 25 4

3 3y+12 3 y+18

Evaluating equal matrices

Holt Algebra 2

4-1 Matrices and Data

x+8 -5 = 38 -5

3 -y 3 4y-10

Holt Algebra 2

4-1 Matrices and Data

x- 1 4 + -4 7 = 5 -1

-3 9 10 -2 6 3

Holt 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 –1 R =

1 4

–2 3

0 4

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

Holt 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 –1 R =

1 4

–2 3

0 4

Holt Algebra 2

4-1 Matrices and Data

Write a short summary to explain how to solve matrix equations.

Holt Algebra 2

4-1 Matrices and Data

• The product of an m n and an n p matrix is an m p matrix.

Holt Algebra 2

4-1 Matrices and Data

Let’s work through this together …

Holt Algebra 2

4-1 Matrices and Data

Let’s work through this together …

Holt Algebra 2

4-1 Matrices and Data

How do you finish the multiplication?

Holt Algebra 2

4-1 Matrices and Data

How do you finish the multiplication?

Holt Algebra 2

4-1 Matrices and Data

Find the products:

1. A2

Holt Algebra 2

4-1 Matrices and Data

Find the products:

1. CB

Holt Algebra 2

4-1 Matrices and Data

Find the products:

1. BA

Holt Algebra 2

4-1 Matrices and Data

Find the products:

1. CD

Holt Algebra 2

4-1 Matrices and Data

Find the products:

1. BA

Holt Algebra 2

4-1 Matrices and Data

The product of two or more matrices is the matrix product. The following rules apply when multiplying matrices.

• The product of an m n and an n p matrix is an m p matrix.

Holt Algebra 2

4-1 Matrices and Data

• The product of an m n and an n p matrix is an m p matrix.

Holt Algebra 2

4-1 Matrices and Data

Tell whether the product is defined. If so, give its dimensions.

Example 1A: Identifying Matrix Products

A3 4 and B4 2; AB

A B AB

3 4 4 2 = 3 2 matrix

Holt Algebra 2

4-1 Matrices and Data

Tell whether the product is defined. If so, give its dimensions.

Example 1B: Identifying Matrix Products

C1 4 and D3 4; CD

C D

1 4 3 4

The inner dimensions are not equal (4 ≠ 3), so the matrix product is not defined.

Holt Algebra 2

4-1 Matrices and Data

Tell whether the product is defined. If so, give its dimensions.

P2 5 Q5 3 R4 3 S4 5

Check It Out! Example 1a

QP

Holt Algebra 2

4-1 Matrices and Data

Tell whether the product is defined. If so, give its dimensions.

P2 5 Q5 3 R4 3 S4 5

Check It Out! Example 1b

SR

Holt Algebra 2

4-1 Matrices and Data

Tell whether the product is defined. If so, give its dimensions.

P2 5 Q5 3 R4 3 S4 5

Check It Out! Example 1c

SQ

Holt Algebra 2

4-1 Matrices and Data

Write a short summary to explain how to find matrix products.

Holt Algebra 2

4-1 Matrices and Data

1. What are the dimensions of A?

2. What is entry D12?

Evaluate if possible.

3. A — C 4. C + D 5. B + D