Matrices

Addition and Subtraction

Competency Goal and Objective

• Competency Goal 1: The learner will use matrices and graphs to model relationships and solve problems.

• Objective 1.01: Use matrices to model and solve problems.• Display and interpret data. • Write and evaluate matrix expressions to solve

problems.

Matrices

• If you have ever used a spreadsheet program on the computer, you have worked with matrices.

• A matrix is a rectangular arrangement of numbers in rows and columns.

• It is usually described by its dimensions, or the number of rows and columns, with the number of rows stated first.

Matrices (cont’d)

• Remember:• A row travels from left

to right• A column travels from

top to bottom.• Each entry in a matrix

is called an element, which is assigned a row number and a column number.

• You try!Identify the position of the circled element in each matrix.

a.

b.

11 15 24

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4 2

0 1

3 6

Making Matrices

• Say we are planning a pizza and video party for a few of our friends and we have to make some decisions about ordering food. We start by calling the pizza houses that deliver in the neighborhood and ask about the price for pizzas, drinks and salads.

Problem (cont’d)

We could take the information we receive and record it in a table, like this:

Vin’s Toni’s Sal’s

Pizza $10.10 $10.86 $10.65

Drinks $1.09 $0.89 $1.05

Salads $3.69 $3.89 $3.85

Problem (cont’d)

Or, we could write it in matrix (or ordered array) form, which simply means writing the numbers in a rectangular array and enclosing them in brackets.

$10. $10. $10.

$1. $0. $1.

$3. $3. $3.

10 86 65

09 89 05

69 89 85

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Square matrices

• A matrix, such as the one we have, is known as a square matrix when the number of rows (m) equals the number of columns (n).

• Our matrix is a 3 x 3 matrix.

Column and Row Matrices

• If we decide to list only the prices of Sal’s offerings, we would have a column matrix of dimensions 3 x 1.

• When we choose to look at the pizza prices alone, they can be represented with a 1 x 3 row matrix.

$10.

$1.

$3.

65

05

85

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$10. $10. $10.10 86 65

Matrix Addition

• Matrix addition is fairly simple! All you have to do is add entry by entry.

• So, if you had to add these two matrices,

0 1 2

9 8 7

6 5 4

3 4 5

LNM

OQPLNM

OQP

Adding Matrices

0 6 1 5 2 4

9 3 8 4 7 5

LNM

OQP

LNM

OQP

6 6 6

12 12 12

Rules for Addition of Matrices

• Remember, that to add matrices they must have the same number of rows and columns!• So a 2 x 3 matrix can not be added to a 2 x 2

matrix.

Subtraction of Matrices

• Subtraction works like addition. It is also entry-wise. So when given:

Find:

1. A – B 2. B – A 3. B - C

A LNM

OQP

1 2 0

0 3 6B

LNM

OQP

0 4 3

9 4 3C

LNM

OQP

0 4

9 4

Matrix Subtraction

A B

LNM

OQP

1 0 2 4 0 3

0 9 3 4 6 3

LNM

OQP

1 6 3

9 7 9

Matrix Subtraction (cont’d)

B A LNM

OQP

0 1 4 2 3 0

9 0 4 3 3 6

LNM

OQP

1 6 3

9 7 9

• Notice, that A-B and B-A are not the same.

• Subtraction is not commutative!

LNM

OQP

LNM

OQP

1 6 3

9 7 9

1 6 3

9 7 9

• Not possible, since the dimensions are not the same.

B C

LNM

OQP

LNM

OQP

0 4 3

9 4 3

0 4

9 4

Your Turn

• Try to solve this problem! Find the values of x and y.

LNM

OQP LNM

OQP LNM

OQP

3

2 0

4 6

3 1

1 7

5 1

x

y

• Did you find that

x + 6 =7 and 2y –3 = -5

x = 1 y = -1

Problems to Try

1. A trendy garment company receives orders from three boutiques. The first boutique orders 25 jackets, 75 shirts and 75 pairs of pants. The second boutique orders 30 jackets, 50 shirts and 50 pairs of pants. The third boutique orders 20 jackets, 40 shirts and 35 pairs of pants. Display this information in a matrix whose rows represent the boutiques and whose columns represent the type of garment ordered. Label the rows and columns of your matrix accordingly.

Problems (cont’d)2. For breakfast Patty had cereal, a medium-sized

banana, a cup of 2% fat milk and a slice of buttered toast. She recorded the following information in her food journal. Cereal: 165 calories, 3 g fat, 33 g carbohydrate, and no cholesterol. Banana: 120 calories, no fat, 26 g carbohydrate and no cholesterol. Milk: 120 calories, 5 g of fat, 11 g carbohydrate, and 15 mg cholesterol. Buttered toast: 125 calories, 6 g fat, 14 g carbohydrate and 18 mg cholesterol. Display this information as a matrix.

Answers

1. Matrix Jackets Shirts Pants

1st Boutique

2nd Boutique

3rd Boutique

2. Answers may vary

25 75 75

30 50 50

20 40 35

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165 120 120 125

3 0 5 6

33 26 11 14

0 0 15 18

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PPPP165 3 33 0

120 0 26 0

120 5 11 15

125 6 14 18

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