Chapter 5.7 Properties of Matrices. Basic Definitions It is necessary to use capital letters to name...

Chapter 5.7

Properties of Matrices

Basic Definitions

It is necessary to use capital letters to name matrices.

Also, subscript notation is often used to name element of a matrix, as in the following Matrix A.

With this notation, the first row, first column element is a11 (read “a-sub-one-one”);

The second row, third column element is a23

and in general the i th row, j th column element is aij

Certain matrices have special names:

because the number of rows is equal to the number of columns.

A matrix with just one row is a row matrix, and a matrix with just one column is a column matrix.

matrix square a ismatrix n nAn

Two matrices are equal if they are the same size and if the corresponding element, position by position are equal.

Using this definition, the matrices

are not equal (even though they contain the same elements and are the same size), since the corresponding elements differ.

35

21 and

53

12

Find the values of the variables for which each statement is true, if possible.

01

yx

qp

12

Find the values of the variables for which each statement is true, if possible.

0

4

1

y

x

Adding MatricesAddition of matrices is defined as follows.

It can be shown that matrix addition satisfies the commutative, associative, closure, identity, and inverse properties.

Find the sum if possible

38

64

98

65


12

3

6

8

5

2


B, A

26

85A if

524

193 B and

Special MatricesA matrix containing only zero elements is called a zero matrix. A zero matrix can be written with any size.

By the additive inverse property, each real number has an additive inverse: if a is a real number, then there is a real number –a such that

a + (-a) = 0 and –a + a = 0

Given matrix A, there is a matrix –A such that

A + -A = 0

The matrix –A has as elements the additive inverses of the elements of A.

For example, if

643

125A

643

125A -then

For example, if

643

125-A A

643

125

000

000

Matrix –A is called the additive inverse, or negative, of matrix A. Every matrix has an additive inverse

Subtracting MatricesThe real number b is subtracted from the real number a, written a – b, by adding a and the additive inverse of b.That is,

a – b = a + (-b)

In practice, the difference of two matrices of the same size is found by subtracting corresponding elements.

Find the difference if possible

85

23

42

65


468 853


B, A

10

52A if

5

3 B and

Multiplying MatricesIn work with matrices, a real number is called a scalar to distinguish it from a matix. The product of a scalar K and a matrix X is the matrix kX, each of whose elements is k times the corresponding element of X.

Find the product

40

325

)4(5)0(5

)3(5)2(5

200

1510

Find the product

1612

3620

4

3

129

2715

We have seen how to multiply a real number (scalar) and a matrix. To find the product of two matrices, such as

405

243A

23

32

46-

B and

first locate row 1 of A

405

243A

23

32

46-

B and

first locate row 1 of Aand column 1 of B,

405

243A

23

32

46-

B and

Multiply corresponding elements, and find the sum of the products.

405

243A

23

32

46-

B and

(-3)(-6) + (4) (2) + (2)(3) 18+ 8 + 6 = 32

The result is the first element for row 1, column 1 in the product matrix.

405

243A

23

32

46-

B and

(-3)(-6) + (4) (2) + (2)(3) 18+ 8 + 6 = 32

Now use row 1 of A and column 2 of B to determine the element in row 1 and column 2 of the product matrix.

405

243A

23

32

46-

B and


405

243A

23

32

46-

B and

(-3)(4) + (4) (3) + (2)(-2) -12 + 12 + -4 = -4


405

243A

23

32

46-

B and


405

243A

23

32

46-

B and

(5)(-6) + (0) (2) + (4)(3) -30 + 0 + 12 = -18


405

243A

23

32

46-

B and


405

243A

23

32

46-

B and

(5)(4) + (0) (3) + (4)(-2) 20 + 0 + -8 = 12

The product matrix can now be written.

405

243

matrix. 22 a ismatrix 23 a and

matrix 32 a ofproduct thehere see As

1218

432

23

32

46-

Can the product AB be calculated?

The following diagram shows that AB can be calculated, because the number of columns of A is equal to the number of rows of B.

23

AMatrix

matrix. 42 a is B while

matrix, 23 a isA Suppos

42

BMatrix

same size

If AB can be calculated, how big is it?

23

AMatrix



42

BMatrix

same size

size of AB = 3 x 4

If BA can be calculated?

BA cannot be calculated?

23

AMatrix



different size

42

BMatrix

Find the product

27

31

1413

2101

)3)(3()1)(1(

Find the product

27

31

1413

2101

8

Find the product

27

31

1413

2101

)1)(3()0)(1(8

Find the product

27

31

1413

2101

38

Find the product

27

31

1413

2101

)4)(3()1)(1(38

Find the product

27

31

1413

2101

1338

Find the product

27

31

1413

2101

)1)(3()2)(1(1338

Find the product

27

31

1413

2101

51338

Find the product

27

31

1413

2101

)3)(2()1)(7(

51338

Find the product

27

31

1413

2101

-8 -3 -13 513

Find the product

27

31

1413

2101

)1)(2()0)(7(13

51338

Find the product

27

31

1413

2101

-8 -3 -13 513 2

Find the product

27

31

1413

2101

)4)(2()1)(7(213

51338

Find the product

27

31

1413

2101

-8 -3 -13 513 2 1

Find the product

27

31

1413

2101

)1)(2()2)(7(1213

51338

Find the product

27

31

1413

2101

-8 -3 -13 513 2 1 12

Find the product

27

31

1413

2101

Since the first matrix is a 2 x 4and the second matrix is a 2 x 2the product can not be found.

Find the product

20

72

52

31

Find the product

20

72

52

31

)0)(3()2)(1(

Find the product

20

72

52

31

2

Find the product

20

72

52

31

)2)(3()7)(1(2

Find the product

20

72

52

31

132

Find the product

20

72

52

31

)0)(5()2)(2(

132

Find the product

20

72

52

31

4

132

Find the product

20

72

52

31

)2)(5()7)(2(4

132

Find the product

20

72

52

31

44

132

Find the product

52

31

20

72

Find the product

)2)(7()1)(2(

52

31

20

72

Find the product

16

52

31

20

72

Find the product

)5)(7()3)(2(16

52

31

20

72

Find the product

2916

52

31

20

72

Find the product

)2)(2()1)(0(

2916

52

31

20

72

Find the product

4

2916

52

31

20

72

Find the product

)5)(2()3)(0(4

2916

52

31

20

72

Find the product

104

2916

52

31

20

72

The products are not commutative.

104

2916

52

31

20

72

A B B A

A contractor builds three kinds of houses, models A, B, and C, with a choice of two styles, colonial or ranch.

Matrix P shows the number of each kind of house the contractor is planning for a new 100-home subdivision.

The amounts for each of the main materials used depend on the style of the house.

20

20

30

20

10

0

C Model

B Model

A ModelColonial Ranch

The amounts are shown in the matrix.

220150

20210

Ranch

ColonialConcrete Lumber Brick Shingles

Concrete is measured here in cubic yards, lumber in 1000 board feet, brick in 1000s, and shingles in 100 square feet.

220150

20210

Ranch

ColonialConcrete Lumber Brick Shingles

25

60

100

20

Shingles

Brick

Lumber

Concrete

Cost per Unit

What is the total cost of materials for all houses of each model?

20

20

30

20

10

0

220150

20210

80

60

60

400

400

600

60

40

30

1200

1100

1500

C Model

B Model

A ModelConcrete Lumber Brick Shingles

What is the total cost of materials for all houses of each model?

80

60

60

400

400

600

60

40

30

1200

1100

1500

25

60

100

20

C Model

B Model

A Model

60800

54700

72900

Cost

How much of each of the four kinds of material must be ordered?

80

60

60

400

400

600

60

40

30

1200

1100

1500

20014001303800

What is the total cost of the materials?

25

60

80

20

20014001303800 188400

Chapter 5.7 Properties of Matrices. Basic Definitions It is necessary to use capital letters to name...

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Transcript of Chapter 5.7 Properties of Matrices. Basic Definitions It is necessary to use capital letters to name...