wednesday, july 15, 2015 eq: what are the similarities and differences between matrices and real...

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Warm UpEvaluate each expression for a = -5, b= 1.3,

and c= -7.

1. a + b

2. b – c

3. a – b + c

4. -4b

Solve each equation.

5. 16 = 2x – 5

6. 3y + 8 = -y - 15

-3.7

8.3

-13.3

-5.2

X= 10.5

Y = -5.75

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Unit 4 Matrices

Lesson 1 Matrix Operations

Essential Question

What are the similarities and differences between matrices and real numbers?

Standards M.ALGII.2.4

Objectives

Vocabulary

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

What we’re learning today:

•What matrices are

•How to add, subtract and multiply matrices.

Unit 4: Lesson 1

Describe how you find your seat in a stadium when you go to a sports game or a concert.

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

The Matrix

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

The Matrix

mnnn

m

m

aaa

aaa

aaa

,2,1,

,22,21,2

,12,11,1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix

(Plural: Matrices)

A matrix is a rectangular array of numbers. Matrices are named using capital letters.

Example:

154

702A

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

The Matrix

mnnn

m

m

aaa

aaa

aaa

,2,1,

,22,21,2

,12,11,1

Row 1

Row 2

Row n

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

The Matrix

mnnn

m

m

aaa

aaa

aaa

,2,1,

,22,21,2

,12,11,1

Column 1 Column 2 Column m

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Dimensions of a Matrix

The dimensions of a matrix are the number of rows and the number of columns in the matrix.

Examples:

154

70223

12

4

3

241

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

The Matrix

mnnn

m

m

aaa

aaa

aaa

,2,1,

,22,21,2

,12,11,1

Entry

(Element)

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Entries

AKA: Matrix Elements

The numbers inside a matrix are called entries or elements. The location or “address” of an entry is the number of the row and column where the entry is located.

Example:

The entry at row 2, column 3 is 1

154

702

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Enter each matrix into your calculator

154

702A

189

315B

01

35C

29

74D

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Equality

Two matrices are equal if and only if they have identical dimensions and all corresponding entries are equal.

Example:

154

702

154

702

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Addition and Subtraction

It is only possible to add or subtract two matrices, if they have identical dimensions. To find the sum, add corresponding entries. To find the difference, subtract corresponding entries.

Examples:

1105

886

059

184

154

702

72

06

3

4Not Possible

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Find the difference without a calculator:

29

74

01

35

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Use your calculator to find each sum or difference:

154

702A

189

315B

01

35C

29

74D

1) A + B2) B + A3) (A + B) + A

4) A + (B + A)

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Properties of Matrix Addition

For matrices A, B and C with identical dimensions:

Commutative: A + B = B + A

Associative: (A + B) + C = A + (B + C)

Additive Identity: A + 0 = A

Additive Inverse: A + -A = 0

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

The Zero Matrix

A matrix with dimension mn whose entries are all 0 is the additive identity for mn matrices.

Examples:nm,0

00

000 2,2

0000 3,1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Scalar Multiplication

A scalar is a real number. To multiply a scalar by a matrix, multiply the scalar by every entry in the matrix.

Example:

31512

2106

154

7023

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Use your calculator to find each product:

154

702A

189

315B

01

35C

29

74D

1) 5·B2) -3·C3) ½·D

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

It is only possible to multiply two matrices when the number of columns in the first matrix is equal to the number of rows in the second matrix.

Example:

31512

2106

154

702Not Possible

31512

2106

27

13Possible

23 · 23

22 · 23

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

The dimensions of the product of two matrices will be the number of rows in the first matrix and the number of columns in the second matrix.

Example:

1533018

66156

31512

2106

27

13

22 · 23 = 23

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

12163

Row 1

Column 1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

1218

Row 1

Column 1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

6Row 1

Column 1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

151036

Row 1

Column 2

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

1506

Row 1

Column 2

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

156

Row 1

Column 2

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

31213156

Row 1

Column 3

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

363156

Row 1

Column 3

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

66156

Row 1

Column 3

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

12267

66156

Row 2

Column 1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

2442

66156

Row 2

Column 1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

18

66156

Row 2

Column 1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

1520718

66156

Row 2

Column 2

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

30018

66156

Row 2

Column 2

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

3018

66156

Row 2

Column 2

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

322173018

66156

Row 2

Column 3

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

61473018

66156

Row 2

Column 3

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

1533018

66156

Row 2

Column 3

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Matrix Multiplication

31512

2106

27

13

1533018

66156

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

32

15

31

42

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Use your calculator to find each product:

154

702A

189

315B

01

35C

29

74D

1) C·D

3) D·A4) A·C5) C·A

2) D·C

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

•Worksheet: Multiplying Matrices #1

Wednesday, April 19, 2023

EQ: What are the similarities and differences between matrices and real numbers?

Worksheet: Multiplying Matrices #2