Unit 2 Day 1

MATRICES

MATRIX OPERATIONS

Warm-UpBasic Matrix Practice

2

1 2

2. 5 4 1

0 5

x

y

x

4 6 51.

6 3 7 3

a

a

1 3 2 2 1 53.

4 0 5 6 4 3

b

c

3 04. 3

4 5

t

q

4 6 51.

6 3 7 3

a

a

1 3 2 2 1 53.

4 0 5 6 4 3

b

c

2 5

7 6 0

a

a

1 4 2 5

4 6 4 8

b

c

Warm-Up ANSWERS! Matrix Addition & Subtraction:

3 04. 3

4 5

t

q

2

1 2

2. 5 4 1

0 5

x

y

x

Warm-Up ANSWERS! Scalar Multiplication:

9 0

12 15

t

q

2

5 10 5

20 5 5

0 25 5

x

y

x

Questions About HW?

Questions About Matrices

Basics HW?

Tonights HW

Is Packet p. 1-2

A heads-up…

The HW is not in order in the

packet, so be sure to refer

to your outline this unit.

Unit 2 Day 1

NOTES Part 1

BASIC MATRIX OPERATIONS

& APPLICATIONS

Remember…Matrix: a rectangular array of numbers or variables used to organize data.

2 4

3 5

1 6

A

Typically, we name matrices with capital letters!

The numbers in the matrix are called elements. There are 6 elements in this matrix.

Remember…Matrix dimensions tell how many ROWS & COLUMNS there are in the matrix.

Dimensions & Notation

A 3 x 2

REMEMBER:RC colaRows by columns

Rows run horizontally & columns run vertically.

The dimensions of two matrices can determine whether or not they may be added, subtracted, or multiplied.

2 4

3 5

1 6

A

A matrix of m rows and n columns is called a matrix with dimensions m x n.

2 3 4

1.) 11

2

3 8 9

2.) 2 5

6 7 8

103.)

7

4.) 3 4

2 X 3 3 X 3

2 X 11 X 2

You Try Examples: Find the dimensions.

Matrix Position

We can give the location of an element in a matrix by naming its row, then its column.

2 4 8 4

3 5 6 3

1 6 2 0

3 is in position ____

6 is in position ____

4 is in position ____

-2 is in position _____

k24

k32

k14

k33

Why use matrices?

Matrix algebra makes mathematical expression and computation easier.

It allows you to get rid of cumbersome notation, concentrate on the concepts involved and understand where your results come from.

Matrices are used to represent real-world datasuch as the habits or traits of populations.

Special MatricesSome matrices have special names

because of what they look like.

a) Row matrix: only has 1 row.

b) Column matrix: only has 1 column.

c) Square matrix: has the same number of rows

and columns.

d) Zero matrix: contains all zeros.

) 3 4Ex

10)

7Ex

10 2)

7 3Ex

) 0 0Ex

Remember…Matrix Addition & Subtraction

You can add or subtract matrices if they have the same

dimensions (same number of rows and columns).

To do this, you add (or subtract) the corresponding

numbers (numbers in the same positions).

If a matrix operation is not possible for a problem, the

solution is called undefined.

Ex:

2 41 2 3

5 00 1 3

1 3

Undefined

Properties of Matrix Addition

Matrix addition IS commutative

A + B = B + A

Matrix addition IS associative

A + (B + C) = (A + B) + C

Remember…Scalar Multiplication

2 4

4 5 0

1 3

To do this, multiply each entry in the matrix by the number outside (called the scalar).

This is like distributing a number to a polynomial.

Example:

8 16

20 0

4 12

Unit 2 Day 1

NOTES Part 2

MATRIX MULTIPLICATION

Matrix Multiplication

Matrix Multiplication is NOT Commutative!

Order matters!

You can multiply matrices only if the number of

columns in the first matrix equals the number

of rows in the second matrix.

2 3

5 6

9 7

2 columns

2 rows1 2 0

3 4 5

3 x 2 2 x 3 = 3 x 3resulting matrix

Matrix Multiplication

Take the numbers in the first row of matrix #1. Multiply

each number by its corresponding number in the first

column of matrix #2. Total these products.

2 3

5 6

9 7

1 2 0

3 4 5

2 1 3 311

Do 1st ● 1st + 2nd ● 2nd +…

The result, 11, goes in row 1, column 1 of the answer. Repeat with row 1, column 2; row 1 column 3; row 2, column 1; ...

Continued on the next slide….

Let’s Try!

2 3

5 6

9 7

1 2 0

3 4 5

Let’s Try Answer!

2 3

5 6

9 7

1 2 0

3 4 5

11 8 15

13 34 30

12 46 35

You Try!2 1 3 9 2

3 4 5 7 6

You Try!Answer

2 1 3 9 2

3 4 5 7 6

2(3) + -1(5) 2(-9) + -1(7) 2(2) + -1(-6)

3(3) + 4(5) 3(-9) + 4(7) 3(2) + 4(-6)

1 25 10

29 1 18

2 x 2 2 x 3

Matrix Multiplication

5

2A

8

45

60

2 1

9 0

10 5

B

2 15

9 02

10 5

BA

Find AB and BA given and

AB is undefined. A 2 x 1 and 3 x 2 cannot be multiplied.

Matrix Multiplication Properties

Matrix multiplication is NOT commutative

AB≠BA

Matrix multiplication IS associative

A(BC)=(AB)C

Matrix multiplication IS distributive

A(B+C)=AB+AC(A+B)C=AC+BC

Packet p. 2 #1Matrix Applications

Two softball teams submit equipment lists for the season.Women’s Team: 12 bats, 45 balls, 15 uniforms

Men’s Team: 15 bats, 38 balls 17 uniforms

Each bat costs $21, each ball costs $4, and each uniform costs $30.

Use matrix multiplication to find the total cost of equipment for each team.

ApplicationAnswer

Practice on Packet p. 2

Matrix Applications Handout #2 and 3

Unit 2 Day 1

NOTES

MATRIX APPLICATIONS

Application Example 1(on power point slide handout)

Ex 1) Last year at Green Hope, there were 214 senior girls, 243 senior guys, 245 junior girls, 233 junior guys, 258 sophomore girls, 288 sophomore guys, and 345 freshman girls, and 303 freshman guys. At Cary, there were 223 senior girls, 252 senior guys, 250 junior girls, 200 junior guys, 260 sophomore girls, 248 sophomore guys, and 352 freshman girls, and 333 freshman guys.

a) Write matrices G and C that represent the population of each school.

b) Add these two together. What does this new matrix represent?

Application Example 2(on power point slide handout)

Last year at Green Hope, there were 214 senior girls, 243 senior guys, 245 junior girls, 233 junior guys, 258 sophomore girls, 288 sophomore guys, and 345 freshman girls, and 303 freshman guys.

Ex 2) Use the matrix of Green Hope students from the last example. If a college claims it is 5 times as large as our school, with students being distributed the same by year and gender, create a new matrix that shows the number of students at the college.

YOU TRY Application: (on power point slide handout)

The A-Plus auto parts store has two outlets, one in Greensboro and one in Charlotte. Among other things, it sells wiper blades, windshield cleaning fluid, and floor mats. The monthly sales of those products are given below.

January Ss G’boro Charlotte

Wiper Blades 20 15

Cleaning Fluid 10 12

Floor Mats 8 4

February Ss G’boro Charlotte

Wiper Blades 23 12

Cleaning Fluid 8 12

Floor Mats 4 5

1) Use matrix arithmetic to calculate the change in sales of each product in each store from January to February. Show your matrix operations. Label your matrices.

YOU TRY Application: (on power point slide handout)The January revenue generated by sales in the Greensboro and Charlotte branches for the A-Plus auto parts store was as follows:

2) Suppose the January revenue at the Greensboro A-Plus store was $140 for wiper blades, $30 for cleaning fluid, and $96 for floor mats. The January revenue for the Charlotte store was $105 for wiper blades, $36 for cleaning fluid, and $48 for floor mats.

a) Write a matrix to represent the data above. Label it.b) If the US dollar was worth $0.76 Canadian dollars at the time, compute the revenue in Canadian dollars using matrix arithmetic. Show your matrix operations. Label your matrices.

Application ANSWERS: The A-Plus auto parts store has two outlets, one in Greensboro and one in Charlotte. Among other things, it sells

wiper blades, windshield cleaning fluid, and floor mats. The monthly sales of those products are given below.

January Ss G’boro Charlotte

Wiper Blades 20 15

Cleaning Fluid 10 12

Floor Mats 8 4

February Ss G’boro Charlotte

Wiper Blades 23 12

Cleaning Fluid 8 12

Floor Mats 4 5

Use matrix arithmetic to calculate the change in sales of each product in each store from January to February. Label your matrices.

23 12 20 15

8 12 10 12

4 5 8 4

Gb Ch Gb Ch

WB

F J CF

FM

3 3

2 0

4 1

Gb Ch

WB

F J CF

FM

Application of Scalar Multiplication ANSWERS: Suppose the January revenue at the Greensboro A-Plus store was $140 for wiper blades, $30 for cleaning fluid, and $96 for floor mats. The January revenue for the Charlotte store was $105 for wiper blades, $36 for cleaning fluid, and $48 for floor mats.

a) Write a matrix to represent the data above. Label it.

b) If the US dollar was worth $0.76 Canadian dollars at the time, compute the revenue in Canadian dollars using matrix arithmetic. Show your matrix operations. Label your matrices.

140 105

. 0.76 0.76 30 36

96 48

Gb Ch

Can Value J

106.4 79.8

22.8 27.36

72.96 36.48

Gb Ch

WB

CF

CM

140 105

30 36

96 48

Gb Ch

WB

J CF

CM

Homework

Packet p. 1 and 2

Extra practice on next slides…

Identify each matrix element.

K =

Organizing Data Into Matrices

3 –1 –8 5

1 8 4 9

8 –4 7 –5

a. k12 b. k32 c. k23 d. k34

Element k12 is –1. Element k32 is –4.

a. K =

k12 is the element in the first

row and second column.

3 –1 –8 5

1 8 4 9

8 –4 7 –5

b. K =

k32 is the element in the third

row and second column.

3 –1 –8 5

1 8 4 9

8 –4 7 –5

Matrix Addition

Examples

2 4 1 0

5 0 2 1

1 3 3 3

Ex 1:

2 41 2 3

5 00 1 3

1 3

Ex 2:

Matrix Addition

Example ANSWERS

2 4 1 0

5 0 2 1

1 3 3 3

Ex 1:

3 4

7 1

2 0

2 41 2 3

5 00 1 3

1 3

Ex 2:

Undefined

The table shows information on ticket sales for

a new movie that is showing at two theaters. Sales are

for children (C) and adults (A).

Adding and Subtracting Matrices

a. Write two 2 2 matrices to represent matinee and evening sales.

Theater C A C A

1 198 350 54 439

2 201 375 58 386

b. Find the combined sales for the two showings.

The table shows information on ticket sales for a new

movie that is showing at two theaters. Sales are for children (C)

and adults (A).

ANSWERS Adding and Subtracting Matrices

a. Write two 2 2 matrices to represent matinee and evening sales.

Theater 1 198 350

Theater 2 201 375

MatineeC A

4-2

Theater 1 54 439

Theater 2 58 386

EveningC A

Theater C A C A

1 198 350 54 439

2 201 375 58 386

(continued)

b. Find the combined sales for the two showings.

198 350

201 375+

54 439

58 386=

198 + 54 350 + 439

201 + 58 375 + 386

=Theater 1 252 789

Theater 2 259 761

C A

4-2

ANSWERS Adding and Subtracting Matrices

Adding & Subtracting MatricesYou can perform matrix addition on matrices with equal dimensions.

a. b.9 0

–4 6 +0 0

0 0

3 –8

–5 1 +–3 8

5 –1

= 9 + 0 0 + 0

–4 + 0 6 + 0=

3 + (–3) –8 + 8

–5 + 5 1 + (–1)

= 9 0

–4 6=

0 0

0 0

4-2