Chapter 11

Section 11.0

Review of Matrices

Matrices

A matrix (despite the glamour of the movie) is a collection of numbers arranged in a rectangle or an array. We use variables like A, B, C, …, [capital letter] to stand for a matrix.

We use what are called double scripted variables with a lower case letter of the matrix to refer to the entries in a matrix. The numbers in the subscript give the position the variable is located in with the first number referring to the row and the second number the column. The dimensions or order of the matrix is given in the form (number of rows) (number of columns). Don't multiply leave it this way!

232221

131211

2170

134

aaa

aaa

411 a 312 a 113 a

021 a 722 a 21

23 a

We name this matrix A

This matrix has 2 rows and 3 columns, or has order or dimension 2 3 "read 2 by 3".

641

0129

375

333231

232221

131211

bbb

bbb

bbb

What do we name this?

What is the entry b21?

What is the entry b12?

What is the variable for 4?

What are the dimensions?

B

9

7

b32

3 3

Types of Matrices

A matrix can have different adjectives that describe it depending on its dimensions.

A matrix is called square matrix if it has the same number of rows and columns. A matrix is called row matrix if it has only one row (i.e. all of the entries are in a single row). A matrix is called a column matrix if it only has one column (i.e. all of the entries are in a single column).

Give the dimensions of each matrix below and determine if it is a square, row or column matrix or if it does not fall in any of the categories.

95

22

03

71

4 2

None

245

8312

106

3 3

Square

5

3

2 1

column

16151413

1211109

8765

4321

4 4

Square

9

1 1Square

rowcolumn

8023

1 4

row

41

113

2 2

Square

0031

0084

2 4

None

65

1 2

row

42931

637104

2 5

None

Matrix Operations

Adding & Subtracting Matrices

The way that matrices are added or subtracted is to add or subtract their corresponding entries. This means that the matrices must be of the same dimensions or order. If they are not we say the two matrices are not the same dimensions we say the matrices are nonconformable.

65

44

28

10)4()7(12

)5(931

4)6(53

107

53

45

412

91

63

BA

620165

19

312

74

DC

The matrices C and D are nonconformable. They can not be added even though they both have 6 entries.

Matrix C is 2 3

Matrix D is 1 6

Multiplication by a Scalar

We can multiply a matrix by a number (sometimes called a scalar) by multiplying each entry in the matrix by the number. This operation can always be done. We say it is always conformable.

14

6

8

)7(2

)3(2

)4(2

7

3

4

22A

43

2

)12()9()7(

)1()2()6(

1297

126

37

31

32

31

31

31

31

31

31

31

31 B

10748615)1(4)2(4)2(3)5(312425343 DC

We can begin to combine more than one operation at a time.

7

21

28

18612

1

3

4

7624373 FE

What you get here is nonconformable since the first matrix is 1 3 and the second matrix is 3 1.

47

23

184520

42910

)6(3)9(5)5(4

)6(7)9(1)5(2

6

9

5

354

712AB

Multiplying Matrices

This is not as obvious an operation as you might think!

It is not as easy as addition or subtraction that you get with the corresponding entries!

What you do is to multiply each entry in a row on the matrix on the left with its corresponding entry in a column of the matrix on the right and add them up.

AB = (rows of matrix A) (columns of matrix B)

Look at the example below:

The matrix A is 2 3 and the matrix B is 3 1. The number of columns for the matrix on the right must be the same as the number of rows for the matrix on the left or else they are nonconformable!

2 3 3 1 2 1

The dimensions of the result are given by the rows of A and columns of B.

7527)1(1)2(4)5(1)0(4)3(1)6(4153

20614

CD

153

206

140

235ED

1 2

2 3

1 3

The matrix E and the matrix D are nonconformable even though they are the same dimensions. The columns and rows do not match up!

2 3

2 3

1

19

)2(2)5(1

)2(2)5(3

2

5

21

23FB

If you multiply a 2 2 matrix by a 2 1 matrix you get another 2 1 matrix!

Identity Matrices

A matrix with the same number of rows and columns is called square. A square matrix with 1's down the top left to bottom right diagonal and 0's off that diagonal is called the identity matrix. They come in different size identity matrices.

10

012I

100

010

001

3I

1000

0100

0010

0001

4I

2 2

3 3

4 4

An identity matrix has the property that if you multiply it either on the right or left by any conformable matrix you get the conformable matrix (i.e. InA = A and AIn = A). The matrix In for matrices acts like the number 1 for numbers.

71

34

7010

0304

71

34

10

01

2221

1211

2221

1211

10

01

aa

aa

aa

aa

6

4

60

04

6

4

10

01

y

x

y

x

y

x

0

0

10

01

Representing Matrix Multiplication

A movie theatre has two prices for movie admission, one for children and one for adult (people over 12). It also charges one rate for a matinee and another for an evening movie given in the table to the right.

Adult Child

Matinee $4 $2

Evening $8 $3

Find the cost of taking 2 adults and 4 children to a matinee movie.

2·4 + 4·2 = 8 + 8 =16

Find the cost of taking 2 adults and 4 children to an evening movie.

2·8 + 4·3 = 16 + 12 =28

A matrix is a rectangular array of numbers notice what we get if we multiply the two matrices below together.

28

16

1216

88

3482

2442

4

2

38

24 Cost of Matinee Movie

Cost of Evening Movie

In other words the matrix multiplication combines all of these calculations into one. This enables you to represent many different calculations at once.