Topic 2 Matrices and System of Linear Equations

7/28/2019 Topic 2 Matrices and System of Linear Equations

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MODULE OUTCOMES:

1. Understand basic mathematical concepts and mathematical

techniques for algebra, calculus and data handling.

2. Apply the mathematical calculations, calculus techniques andstatistical methods in industry.


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LEARNING OUTCOMES

At the end of this topic, student should be able to:

Definition of matrix.

Identify the different types of matrices such as rectangular, column,

row, square , zero / null , diagonal, scalar, upper triangular, lower

triangular and identity matrices. Solve the equality of matrices.

Perform operations on matrices such as addition, subtraction, scalar

multiplication of two matrices.

Identify Transpose of Matrix.

Define the determinant of matrix and find the determinant of 2 x 2

and 3 x 3 matrix.

Write a system of linear equations

Solve the system of linear equations by using Cramers Rule.


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Introduction To Matrix Definition Of Matrix

Types Of Matrix

Algebraic OperationsAddition Subtraction

Multiplication

Determinant Of Matrices System of Linear Equations with Two and

Three Variables by Using Cramers Rule

MARTICES


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Definition of matrices


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Matrix - a rectangular array of

variables or constants in horizontal

rows and vertical columns enclosed inbrackets.

Element - each value in a matrix; either

a number or a constant.

Dimension - number of rows by

number of columns of a matrix.

**A matrix is named by its dimensions.


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row column
http://en.wikipedia.org/wiki/File:Matrix.svg


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Row = 1

Column = 2

Order = 1 X 2

Row = 2

Column = 2

Order = 2 X 2

Order of the matrix


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Examples: Find the dimensions or

order of each matrix.

1. A =

2 1

0 5

4 8

2. B =

1

2

3

4

0 5 3 13. C =

2 0 9 6

Order: 3x2 Order : 4x1

Order : 2x4


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4. A = Order of a matrix = row x column

= (m x n)

Order of A matrix = 3 x 3

Element of A matrix = 1,2,3,4,5,6,7,8,9

987

654

321

7

4

1

31

21

11

a

a

a

8

5

2

32

22

12

a

a

a

9

6

3

33

23

13

a

a

a

Elements of matrix A:


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Types Of Matrices

Equality of Matrices

Introduction


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67237

89511

36402

TYPES OF MATRICES

3410

200

318

0759

00

00

6

0

7

9

3 x 3

3 x 5

2 x 24 x 1

1 x 4

(zero

Matrix)(Column

matrix)

(squarematrix)

(called a row

matrix)

(rectangular

matrix)


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TYPES OF MATRICES

3 x 3

3 x 3

3 x 33 x 3

3 x 3

(lower triangular

matrix)(identity matrix)

(diagonalmatrix)

(scalar matrix)

(upper triangular

matrix)


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Equality of Matrices

Two matrices are equal if they

have the same order and sameentries.


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Example


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Exercises

1. Find the value of x and y for the following: [MO1](a)

(b)

(c)

(d)

43

50

3

5

y

x

102

43

52

3

yxyx

y

x

32

21

92

1

z

y

y

x

x 4

32

2


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Algebraic Operations

Addition/Subtraction

Additions or subtractions of matrices canbe done if they have the same dimensionswhereby the two matrices must have the

same number of rows and the samenumber of columns.

When two matrices are added orsubtracted then the order of matrixshould be the same.


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310

221A

412

403B

2BA

IfA andB are both mn matrices then the sum ofA andB,

denotedA +B, is a matrix obtained by adding correspondingelements ofA andB.

add these

310

221A

412

403B

22BA

add these

310

221A

412

403B

622BA

add these

310

221A

412

403B

2

622BA

add these

310

221A

412

403B

02

622BA

add these

310

221A

412

403B

102

622BA

add these


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Example :

A + B = C

1210

71

87

50

43

21


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Multiplication

Scalar Multiplication

Example :

A =

2A =

=

47

71

47

712

814

142


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To multiply matrices A and B

look at their dimensions

pnnm MUST BE SAME

SIZE OF PRODUCT

If the number of columns ofA does notequal the number of rows ofB then the

product AB is undefined.


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Multiplication of Two Matrices

Necessary condition for matrix multiplication

Column of first matrix should be equal to the rowof the second of matrix.

Example :

00

21

04

32

00240014

03220312

84

42

x

+


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The multiplication of matrices is easier shown than put

into words. You multiply the rows of the first matrix

with the columns of the second adding products

140123A

13

31

42

B

Find AB

First we multiply across the first row and down the

first column adding products. We put the answer in

the first row, first column of the answer.

23 1223 5311223

Fi d AB


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140

123

A

13

31

42

B

Find AB

We multiplied across first row and down first column

so we put the answer in the first row, first column.

5AB

Now we multiply across the first row and down the second

column and well put the answer in the first row, second

column.

43 3243 7113243

75AB

Now we multiply across the second row and down the first

column and well put the answer in the second row, first

column.

20 1420 1311420

1

75AB

Now we multiply across the second row and down the

second column and well put the answer in the second row,

second column.

40 3440 11113440

111

75AB

Notice the sizes ofA andB and the size of the productAB.


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6

BA

126

BA

2126

BA

3

2126

BA

143

2126

BA

4143

2126

BA

9

4143

2126

BA

109

4143

2126

BA

4109

4143

2126

BA

Now lets look at the product BA.

13

31

42

B

140

123

A

BAAB

2332across first row as

we go down first

column:

60432

across first row as

we go down

second column:

124422

across first row as

we go down third

column:

21412

across second row

as we go down

first column:

30331

across second row

as we go down

second column:

144321

across second row

as we go down

third column:

41311

across third row

as we go down

first column:

90133

across third row

as we go down

second column:

104123

across third row

as we go down

third column:

41113

Completely different than AB!

Commuter's Beware!


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Computation: A x B = C

A

2 3

1 1

1 0

and B 1 1 1

1 0 2

[3 x 2] [2 x 3]A and B can be multiplied

111

312

825

12*01*110*01*111*01*1

32*11*110*11*121*11*1

82*31*220*31*251*31*2

C

[3 x 3]


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Computation: A x B = C

111

312

825

12*01*110*01*111*01*1

32*11*110*11*121*11*1

82*31*220*31*251*31*2

C

A

2 3

1 1

1 0

and B 1 1 1

1 0 2

[3 x 2] [2 x 3]

[3 x 3]

Result is 3 x 3


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Exercises

1. A = B = C =

Find: [MO2](a) A + C

(b) CA

(c) 3A2C

(d) A + 2C

(e) AB

201

043

431

510

216

123

512


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Example:

A =

03

21

02

31TA


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Example


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Try this:


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Determinant of Matrix

The determinant of matrix is a unique realnumber for every square matrix. The

determinant of a square matrix is denoted by

Det A or .

Determinant of Matrix 2 x 2

Let us consider a 2 x 2 matrix :

2221

1211

aa

aaA

21122211aaaaA

A

-


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Example :

Find the value of the determinant for matrix A.

Solution :

59

73A

)97()53( A

48


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Determinant for Matrix 3 x 3

Let us consider a 3 x 3 matrix :

333231

232221

131211

aaa

aaa

aaa

A

3231

2221

13

3331

2321

12

3332

2322

11aa

aaa

aa

aaa

aa

aaaA

Should remember that || is a sign of

the determinant


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Example

512

214

321

A A


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Continue :

)24(3)420(2)25(1

35


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Systems of Linear Equations [MO2]A system of linear equations is a collection of two @ morelinear equations, each containing one or more variables.

The following is a system of three equations containing threevariables.

Using a matrix notation, we can write this system in simplifiedform.

This is called the augmented matrix of the system.

5322

16234

12

zyx

zyx

yx

516

3

322234

011


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Matrix equation

5322

16234

12

zyx

zyx

yx


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Exercise

Write the augmented matrix of each system. [MO2]

(a)

(b)

532

643

yx

yx

082

0102

yx

zxzyx


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Solving a system using Cramers Rule [MO2]

Consider the pair of simultaneous equations

Let the matrix of coefficient be , that is

Therefore by using Cramers Rule

for 2 x 2 Matrix

A

qdycx

pbyax

dc

baA

A

dqbp

X

A

qcpa

Y


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Example:

Solve the system by using Cramers Rule

8x+5y=22x-4y=-10


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Continue:

By using Cramers Rule:

42

410

52

x

42

102

28

y

142

42

42

)50(8

42

410

52

x 2

42

84

42

480

42

102

28

y

Therefore, x = -1, and y = 2


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Example:Solve the system 3x - 2y +z= 9

x + 2y - 2z= -5x +y - 4z= -2

Cramers Rule

x

9 2 1

5 2 2

2 1

4

3 2 1

1 2 2

1 1 4

2323

1 y

3 9 1

1 5 2

1

2

4

3 2 1

1 2 2

1 1 4

6923

3


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Example, continued: 3x - 2y +z= 9

x + 2y - 2z= -5x +y - 4z= -2

Cramers Rule

z

3 2 9

1 2 5

1 1 2

3 2 11 2 2

1 1 4

0

23 0

The solution is

(1, -3, 0)


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Three shops A, B, and C sells three

grades of T-shirts with the top grade

providing a profit of RMxa piece, and the

moderate and low grades providing profitsof RMyand RMza piece respectively. On

a certain day, the number of the three

grades sold and the total profit of each ofthe types are shown in the table below:

Example:


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Grades of T-shirtsProfit

Shop

Low

Moderate

Top

A 10 20 30 RM260B 20 40 50 RM460C 30 50 60 RM570

Use the Cramers Rule to find x, y, and z.


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Solution:


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Continue:


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Exercise:

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Topic 2 Matrices and System of Linear Equations