Review

Solve the system of equations.

1

2

22

23

zyx

zyx

zyx

2, -1, 1

Matrices

2

Definition

• Some Words: One: Matrix

More than one: Matrices • Definition: In Mathematics, matrices are used

to store information.• This information is written in a rectangular

arrangement of rows and columns.• Each entry, or element, of a matrix has a

precise position and meaning.3

Example

• Food shopping online: people go online to shop for items and have them delivered to their homes.

• Cartons of eggs, bread, packets of vegetables, bags of rice, packets of fish were ordered online and the people left their address for delivery.

• A selection of orders may look like this:

4

ExampleOrder

Address

Carton of eggs

bread vegetables rice fish

10 Kros

Road

0 2 2 2 1

15 Usmar St

0 2 1 1 3

17 High St 1 2 1 0 0

22 Ofar Rd.

4 0 0 1 35

Example

• The dispatch people will be interested in the numbers:

This is a 4 by 5 matrix

0 2 2 2 1

0 2 1 1 3

1 2 1 0 0

4 0 0 1 3

4 rows

5 columns 6

Definition

A matrix is defined by its order which is always number of rows by number of columns

7

R X C

2 rows

3 columns

2 X 3 matrix

2 5 8

1 6 1

Exercise• Consider the network below showing the

roads connecting four towns and the distances, in km, along each road.

8

A

14

C

D

B5

10

8

12

16

(i) Write down the information in matrix form. (ii) What is the order of the matrix?

Solution(i) This information could be put into a table:

9

km

A B C D

A 0 5 14 12

B 5 0 10 16

C 14 10 0 8

D 12 16 8 0

to

from

Solutionand then into a matrix:

10

0 5 14 12

5 0 10 16

14 10 0 8

12 16 8 0

(ii) order: R X C = 4 X 4 matrix.This is called a square matrix.

Definition

A square matrix has the same number of rows as columns. Its order is of the form M x M.

Examples:

11

1 0

0 1

2 X 2 square matrix

2 0 6

3 5 18

7 8 3

3 X 3 square matrix

DefinitionThe transpose of a matrix M, called MT, is found by interchanging the rows and columns.

Example: M =

12

2 3

7 9

2

3

7

9

rowrow

column

Definition

Equal Matrices: Two matrices are equal if theircorresponding entries (elements) are equal.

Example: If

13

a b

c d

10 2

4 8

a = 10

c = 4 d = 8

b = -2

=

DefinitionEntries, or elements, of a matrix are named according to their position in the matrix. The row is named first and the column second.

Example: entry a23 is the element on row 2, column 3.

Example: here are the entries for a 2 x 2 matrix.

14

11 12

21 22

a a

a a

ExampleIn the following matrix, name the position of the colored entry.(i)

15

1-752

Remember: row firsta2

Column second

row 2

column 1The entry is a21

Example

In the following matrix, name the position of the colored entry.

(ii)

16

c d e f

o p q r

row 1, column 3

The entry is a13

Example• In the following matrices, identify the value of

the entry for the given position.

17

7 8

2 1

3 5

7 5 3 0

10 9 0 2

1 0 5 11

a32

a24

row 3, column 2

= 5

row 2, column 4= 2

College Kaitlin wants to attend one of three Iowauniversities next year. She has gathered informationabout tuition (T), room and board (R/B), andenrollment (E) for the universities. Use a matrix toorganize the information. Which university’s total cost is lowest?

Iowa State University:T - $3132 R/B - $4432 E - 26,845

University of Iowa:T - $3204 R/B - $4597 E - 28,311

University of Northern Iowa:T - $3130 R/B - $4149 E - 14,106

Organize the data into labeled columns and rows.

ISU

UI

UNI

T R/B E

Answer: The University of Northern Iowa has the lowest total cost.

Dining Out Justin is going out for lunch. The information he has gathered from the two fast-food restaurants is listed below. Use a matrix to organize the information. When is each restaurant’s total cost less expensive?

Burger Complex Lunch Express

Hamburger Meal $3.39 Hamburger Meal $3.49

Cheeseburger Meal $3.59 Cheeseburger Meal $3.79

Chicken Sandwich Meal $4.99 Chicken Sandwich Meal $4.89

The Burger Complex has the best price for hamburgers and cheeseburgers. Lunch Express has the best price for chicken sandwiches.

Answer:

Burger Complex

Lunch Express

HamburgerMeal

Cheese-burgerMeal

ChickenSandwich

Meal

State the dimensions of matrix G if

4 columns

2 rows

Answer: Since matrix G has 2 rows and 4 columns, the dimensions of

matrix G are 2 4.

State the dimensions of matrix G if

Answer: 3 2

Solve for x and y.

Since the matrices are equal, the corresponding elements are equal. When you write the sentences to solve this equation, two linear equations are formed.

Second equation

This system can be solved using substitution.

Substitute 3x – 2 for y.

Distributive Property

Add 4 to each side.

Divide each side by 7.

To find the value for y, substitute 1 for x in either equation.

Substitute 1 for x.

First equation

Simplify.

Answer: The solution is (1, 1).

Solve for x and y.

Answer: (2, 5)

Review

State the dimensions of each matrix. Then identify the position of the circled element in each matrix.

28

Review

Solve the equation.

29

935332 yzx

12

15

5

4

y

xx

35

5

4

512 x

yyx

yx