Unit 2 Day 1

MATRICES

MATRIX OPERATIONS

About Matrices

A matrix is a rectangular arrangement of numbers in rows and columns. Rows run horizontally and columns run vertically.

The dimensions of a matrix are stated “m x n” where ‘m’ is the number of rowsand ‘n’ is the number of columns.

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

Examples: Find the dimensions.

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 data such 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

Matrix Addition

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).

2 4 1 0

5 0 2 1

1 3 3 3

Ex:

Matrix Addition

2 4 1 0

5 0 2 1

1 3 3 3

Example Answer:

3 4

7 1

2 0

4 1 6 51.

6 3 7 3

1 3 2 2 1 52.

4 0 5 6 4 3

Practice

4 1 6 51.

6 3 7 3

1 3 2 2 1 52.

4 0 5 6 4 3

2 6

13 0

1 4 7

2 4 8

Practice Answers

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 the two stores for two months 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.

Properties of Matrix Addition

Matrix addition IS commutative

A + B = B + A

Matrix addition IS associative

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

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:

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.

8 16

20 0

4 12

Example Answer:

3 01. 3

4 5

2

1 2

2. 5 4 1

0 5

x

y

x

You Try! Scalar Multiplication:

3 01. 3

4 5

9 0

12 15

2

1 2

2. 5 4 1

0 5

x

y

x

2

5 10 5

20 5 5

0 25 5

x

y

x

Scalar Multiplication Answers:

Scalar Multiplication Application: The January revenue generated by sales in the Greensboro and Charlotte branches for the A-Plus auto parts store was as follows:

January Revenue Greensboro Charlotte

Wiper Blades 140 105

Cleaning Fluid 30 36

Floor Mats 96 48

If the US dollar was worth $0.76 Canadian dollars at the time, compute the revenue in Canadian dollars using matrix arithmetic.

140 105

0.76 30 36

96 48

106.4 79.8

22.8 27.36

72.96 36.48

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

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 not possible. 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

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

Solving Systems of Equations with Matrices

Put these notes on the BACK of your Notes handout!

Matrix Equation

6

8

21

45

y

x

A linear system can be written as a matrix equation AX=B

Coefficient matrix Variable

matrix

Constant matrix

5 4 8

1 2 6

x y

x y

Suppose ax = b

How do you solve for x?

We cannot divide matrices, but we can multiply by the inverse.

AX = BA-1 A-1

X = A-1B

X = A-1B

Solving Matrix Equations

Ex. 2 Solve using matrices.

3 4 5

2 10

x y

x y

x = -7

y = -4

10

5

12

43

y

x

A B

X = A-1B

AX = B

(-7, -4)

Ex. 3 Solve using matrices

2 3 1

3 3 1

2 4 2

x y z

x y z

x y z

x = 2

y = -1

z = -2

(2, -1, -2)

You Try! Solve using matrices

x y z

x y

x y z

3 5 15

2 1

9 8 4 12

Ex. 5

7 3 11

14 4 2

x y

x y

Ex. 4

You Try Answers! Ex. 4 Solve using matrices

7 3 11

14 4 2

x y

x y

x = 5/7

y = 2

(5/7, 2)

You Try Answers! Ex. 5 Solve using matrices

x y z

x y

x y z

3 5 15

2 1

9 8 4 12 x = 4

y = -7

z = 2

(4, -7, 2)

Example

The sum of three numbers is 12. The first is five times the second and the sum of the first and third is 9. Find the numbers.

Let x = first number, y = second number, z = third number

12x y z

5x y

9x z

12

5 0

9

x y z

x y

x z

(15,3, 6)

You try:

1. My mom has three brothers. Together, their ages total 108. The youngest is 8 years less than the oldest. The middle one is four years older than the youngest. How old is each brother?

2. I have nickels, dimes, and quarters in my piggy bank. When I totaled it up last weekend, I had $12.60. I remember I had 110 coins, and that there were only two more dimes than quarters. How many of each type did I have?

You try:

1. My mom has three brothers. Together, their ages total 108. The youngest is 8 years less than the oldest. The middle one is four years older than the youngest. How old is each brother?

2. I have nickels, dimes, and quarters in my piggy bank. When I totaled it up last weekend, I had $12.60. I remember I had 110 coins, and that there were only two more dimes than quarters. How many of each type did I have?

108

8

4

x y z

x z

y x

110

.05 .10 .25 12.60

2

n d q

n d q

d q

(32, 36, 40)

52

30

28

3. Janet is spending the allowance she has saved on clothes. If she buys 3 shirts, 2 skirts, and 4 pairs of jeans, she will spend $292. If she buys 4 shirts, 1 skirt, and 3 pairs of jeans, she will spend $252. If jeans cost $4 more than skirts, find the price of each item.

4. At Morgan’s Fine Cuisine, meals are served a la carte. That is, each item on the menu is priced separately. Jackie and Ted Paris went to celebrate their anniversary. Jackie ordered prime rib, 2 side dishes, and a roll. Ted ordered prime rib, 3 side dishes, and 2 rolls. Jackie’s meal cost $36 while Ted’s cost $44. If the prime rib is three times as expensive as a side dish, what is the cost of each item?

5. The measure of the largest angle of a triangle is twice the measure of the smallest angle. The sum of the smallest angle and the largest angle is twice the other angle. Find the measure of each angle.

You try!

3. Janet is spending the allowance she has saved on clothes. If she buys 3 shirts, 2 skirts, and 4 pairs of jeans, she will spend $292. If she buys 4 shirts, 1 skirt, and 3 pairs of jeans, she will spend $252. If jeans cost $4 more than skirts, find the price of each item.

(28, 32, 36)

You try Answers:

4. At Morgan’s Fine Cuisine, meals are served a la carte. That is, each item on the menu is priced separately. Jackie and Ted Paris went to celebrate their anniversary. Jackie ordered prime rib, 2 side dishes, and a roll. Ted ordered prime rib, 3 side dishes, and 2 rolls. Jackie’s meal cost $36 while Ted’s cost $44. If the prime rib is three times as expensive as a side dish, what is the cost of each item?

($21, $7, $1)

You try Answers:

5. The measure of the largest angle of a triangle is twice the measure of the smallest angle. The sum of the smallest angle and the largest angle is twice the other angle. Find the measure of each angle.

(40, 60, 80)

You try Answers:

Homework

Packet p. 1