unit4 - matrices

7/29/2019 UNIT4 - Matrices

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TOPIC 4 MATRICES

MATRICES

4.0 INTRODUCTION

Matrices are sets of numbers that are arranged in rectangular forms. It is a rectangular

array of numbers. These numbers are arranged inside a round or square bracket. Look at the

examples shown below.

( )8324

45

2

325

793

32

41

69

16

It is important to study about the fundamentals of matrices first and get a good introduction to how

to apply simple algebra operations on matrices. This can help in solving engineering problems.

For example, you can use matrices to solve systems of linear simultaneous equations.

4.1 FUNDAMENTALS OF MATRIX

The array of numbers inside a matrix is called the elements of the matrix. These

numbers are arranged in rows and columns.

Rows are the horizontally arranged elements of the matrix

For example, the shaded region in the matrix below is the second row of the matrix.

252

620

430

441

Columns are the vertically arranged elements of the matrix.

For example, the shaded region in the matrix below is the second column of the matrix.

252620

430

441

Note: It is common practice to use capital letters like A to represent a matrix, and small letters to

represent the elements.

4.2 SIZE OF A MATRIX

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The size of a matrix is the number of rows and columns that it has. If a matrix has 3

rows and 4 columns, then its size is 3 x 4. Lets look at the following matrix.

A =

6935

1038

7441

How many rows and columns do you see? Do you agree that the size of matrix A is 3 x 4?

Example 1:

State the size of the following matrix.

047

933

663

194

Solution:

There are 4 rows and 3 columns. Therefore, the size of this matrix is .

For a matrix A of size 3 x 4, you can use the notation A34 to represent the matrix. In general, any

matrix can be represented by the notation matrix Aik with i = 1, 2, 3, ., and k = 1, 2, 3,

The first subscript, i, represents the rows and the second subscript, j, represents the columns.

ACTIVITY 4a

1. State the size of each of the following matrices:

a. ( )32 b.

5

4

3

c.

672

413

201

d.

75

91

23

2. Referring to matrix B =

271

480

353

, state the element at:

a. b23

b. b21

c. b31

4.3 TYPES OF MATRIX

4.3.1 Square Matrix

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A square matrix is a matrix where the number of rows is equal to the number of

columns. The following examples are square matrices.

5343

271

480

353

15491

317210

231254

91623

4.3.2 Diagonal Matrix

If all the elements of a square matrix consist of zeros except the diagonal, then this

matrix is called a diagonal matrix. The following examples are diagonal matrices.

33

22

11

00

00

00

a

a

a

5003

200

080

003

4.3.3 Identity Matrix

If all the elements of a diagonal matrix consist of the value 1, then the matrix is

an identity matrix. The following examples are identity matrices.

I =

100

010

001

I =

10

01 I =

1000

0100

0010

0001

An identity matrix is special because when you multiply a matrix with it or when

you multiply it with a matrix, the matrix does not change. For examples:

AI = IA = A, IB = BI = B

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4.3.4 Transpose of a Matrix

When you interchange the rows of a matrix with its columns, you would have

converted a matrix Amn to another matrix Anm. In other words, a matrix of size m x n will

now be of size n x m. This new matrix is called the transpose of a matrix. The symbol for a

transpose of a matrix A is AT. Lets look at the following example.

If A =

3231

2221

1211

aa

aa

aa

, then AT =

232221

131211

aaa

aaa.

If A =

106

612

002

, then AT =

160

010

622

.

The transpose of a transpose is the original matrix. (AT)T = A

Some important properties relating to transpose are:

(AB)T = BTAT

(ABCZ)T = ZT..BTAT

(A + B)T = AT + BT

4.3.5 Symmetric Matrix

If the transpose of a matrix is the same as the original matrix, then it is called a

symmetric matrix. Therefore, if A = AT, then A is a symmetric matrix. The following

examples are symmetric matrices.

453

502

321

927

326

761

Activity 4b:

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1. Determine the types of the following matrices:

A =

50

03B =

23

42C =

001

010

100

D =

10

01

2. Determine whether the following matrices are symmetric or not.

a)

632

321b)

305

028

681

c)

453

502

321

d)

937

326

761

4.4 MATRIX ADDITION AND SUBTRACTION

Matrix addition / subtraction can only be performed on matrices that have the same size.

The result of a matrix addition / subtraction is a new matrix that is of the same size. All we need

to do is to match the elements that are at the same position in their matrices.

Example 2:

Given that

=

3517

2331

10413

A and

=1031

0314

0852

B

Determine A + B and A B.

Solution:

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4.5 MATRIX MULTIPLICATION

In order to be able to multiply two matrices AB, we have to ensure that the number of

columns in matrix A is the same as the number of rows in matrix B. That means we can multiply

matrix Amn with matrix Bnk because matrix A has n columns and matrix B has n rows too. The

result is a new matrix that has m rows and k columns.

Example 3:

Find the multiplication of

=

10

25A and

=

11

73B .

Solution:

Example 4:

Find the product of matrix A and B, given that

A =

453

502

321

and B =

305

028

681

.

Solution:

.

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ACTIVITY 4c

1. Based on the following matrices,

A =

59

73 B =

53

43 C =

1254 D =

7524

Determine:

a. A + B

b. A C

c. D + (B A)

d. B + C

2. Given that A =

14

12and B =

32

10, Find AB and BA.

3. If P =

271

480

353

, Q =

135

797

531

, R =

941

302and S =

53

16

24

. Find

the product of: a) PQ b) P2 c) QI d) QR e) RS f) SQ

4. Given A =

43

02, B =

31

12and C =

031

214. Find the value for:

a) ( A + B )T b) ( AB )T c) CT BT

5. Find the values of x and y for the followings:

a)

y

x

1

2+

y

x

2

32=

83

59

b)

10

71051

y

x

001

051212

=

001

051265

4.6 DETERMINANT OF A MATRIX

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The determinant of a square matrix is a special number that can be calculated from the

matrix. It is used to represent the real-value of the matrix which can be used to solve simple

algebra problems later on. The symbol for the determinant of matrix A is det(A) or A.

For a matrix of size 2 x 2, the method to find the determinant is:

If A =

dc

ba,

then, det(A) = A =dc

ba= (ad bc).

For a matrix of size 3 x 3, the method to find the determinant is:

If A =

333231

232221

131211

aaa

aaa

aaa

,

then A=3231

2221

31

3313

2312

21

3332

2322

11aa

aaa

aa

aaa

aa

aaa +

therefore, A= ( ) ( ) ( )312232211331233321123223332211 aaaaaaaaaaaaaaa +

Example 5:

If A =

87

65, determine det(A).

Solution:

Example 6:

Determine the determinant of matrix

212

034

231

Solution:

Method 1 : Matrix minor (Matriks Minor)

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Method 2 : Cross Multiplication (Darab silang)

ACTIVITY 4d

1. Determine the determinants for the following 2x2 matrices:

a)

124

136b)

35

83c)

36

24

2. Given that A =

724

432

612

, B =

127401

351

and C =

052

640

241

Determine:

a) A b) B c) C

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4.7 MINOR OF A MATRIX

The Minor of a matrix is a new matrix where all the elements are determinants. Each

determinant is calculated by removing a row and a column from the original matrix. For example,

in order to determine the element at position ij, you will have to remove row i and columnj from

the matrix. Next, you calculate the determinant of what is left.

If A =

333231

232221

131211

aaa

aaa

aaa

, then Minor of A =

333231

232221

131211

MMM

MMM

MMM

Where:

3332

2322

11aa

aa

M = by removing row 1 and column 1 from A

3331

2321

12aa

aaM = by removing row 1 and column 2 from A

3231

1211

23aa

aaM = by removing row 2 and column 3 from A

and so on

Example 7:

If A =

864

297

531

, determine Minor of A.

Solution:

The elements are:

86

2911 =M = 9(8) 2(6) = 60 84

2712 =M = 7(8) 2(4) = 48

64

9713 =M = 7(6) 9(4) = 6

86

5321=M = 3(8) 5(6) = - 6

84

5122=M = 1(8) 5(4) = -12

64

3123 =M = 1(6) 3(4) = -6

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29

5331=M = 3(2) 5(9) = -30

27

5132 =M = 1(2) 5(7) = -33

97

3133 =M = 1(9) 3(7) = -12

Therefore, Minor of A =

123330

6126

64860

ACTIVITY 4e

Find the minor for the following matrices:

i) A =

724432

612

ii) B =

052640

241

4.8 COFACTOR OF A MATRIX

Once you have found the Minor of a matrix, you can easily determine the Cofactor of the

matrix. All the hard work is already done when you determine the Minor of a matrix. All you need

to do now is to multiply each element of the Minor of the matrix with a factor ( ) ji+1 and the

Cofactor is done.

Lets look at the following descriptions:

If A =

333231

232221

131211

aaa

aaa

aaa

and Minor A =

333231

232221

131211

MMM

MMM

MMM

Then, Cofactor of matrix A =

333231

232221

131211

KKK

KKK

KKK

where ijji

ij MK+= )1( .

Therefore,

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Cofactor of matrix A =

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

33

6

32

5

31

4

23

5

22

4

21

3

13

4

12

3

11

2

111

111

111

MMM

MMM

MMM

Example 8:

If A =

864

297

531

, determine the Cofactor of matrix A.

Solution:

First, the Minor of matrix A =

123330

6126

64860

Next, multiply each element by its factor ( ) ji+1

Therefore, you get the

Cofactor of matrix A =

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

121331301

6112161

61481601

654

543

432

=

123330

6126

64860

ACTIVITY 4f

Find the cofactor for the following matrices:

i) A =

135

797

531

ii) B =

321

431

422

4.9 ADJOINT OF MATRIX

For a square matrix A with n x n, you can find the adjoint of a matrix when transposing

the cofactors of a matrix A. In this case, for matrix A,

Adjoint of a matrix A, written as Adj(A) = KT where K are the cofactor for A.

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Then, if A =

333231

232221

131211

aaa

aaa

aaa

and Minor A =

333231

232221

131211

MMM

MMM

MMM

And Cofactor matrix A =

333231

232221

131211

KKK

KKK

KKK

Then adjoint matrix A, Adj(A) = KT =

332313

322212

132111

KKK

KKK

KKK

Example 9:

If A =

864

297

531

, determine the adjoint for matrix A.

Solution:

Inputs from Example 9 and Example 10, you will find minor for A =

1233306126

64860

And matrix of cofactor from A =

123330

6126

64860

Then, adjoint of matrix A, Adj(A) = AT =

1266

331248

30660

AKTIVITY 4g

Find the adjoin matrix for the following matrices:

i) A =

421430

322

ii) B =

254

132

321

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4.10 INVERSE MATRIX

If A is a square matrix, A -1 is called the inverse matrix of A. Then AA-1 = I (Identity matrix).

ACTIVITY 4h

Calculate the inverse matrix for each of the matrix below:

i) A =

141

112

111

ii) B =

339

123

253

4.11 SYSTEMS OF LINEAR EQUATIONS

You have understood the different types of matrices and their operations. You have also

learned to determine the inverse of a matrix. Using the inverse of a matrix, you can solve

simultaneous equations using Cramer Rule and the Inverse method.

4.11.1 THE INVERSE METHOD

Consider the system of equation:

1131211 bzayaxa =++

AdjA

A

A11

=

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2232221 bzayaxa =++

3333231 bzayaxa =++

You have to write it in a matrix form, Ac = b.

333231

232221

131211

aaa

aaa

aaa

z

y

x

=

3

2

1

b

b

b

Where matrix A =

333231

232221

131211

aaa

aaa

aaa

, matrix c =

z

y

x

and matrix b =

3

2

1

b

b

b

. Therefore c = A-

1

b.

To determine c =

z

y

x

, you have to multiply the inverse of A,A-1 to b.

Example 10:

Solve the linear system:

x + 3y + 3z = 4

2x 3y 2z = 2

3x + y + 2z = 5

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Solution:

Rewrite in the form of matrix, Ac = b:

213

232

331

z

y

x

=

5

2

4

Determine A-1 for matrix A,

Determinant for A, A = -1 and Minor of A =

983

873

11104

Next, we find the cofactor of Awhich is

983

873

11104

Now the adjoin of A = AT is

9811

8710

334

Finally, the inverse of A is

9811

8710

334

Therefore

z

y

x

=

9811

8710

334

5

2

4

z

y

x

=

1514

7

x = 7, y = 14, z = -15

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4.11.2 CRAMERS RULE

Another method of solving systems of linear equations is using Cramers Rule where you have to

calculate the determinants of the matrices involved.

For a matrix equation,

333231

232221

131211

aaa

aaa

aaa

3

2

1

x

x

x

=

3

2

1

b

b

b

Let A =

333231

232221

131211

aaa

aaa

aaa

.

You get A1, by substituting

3

2

1

b

b

b

into column 1 of matrix A.

Therefore A1 =

33323

23222

13121

aab

aab

aab

Using the same method for A2 =

33331

23221

13111

aba

aba

aba

and A3=

33231

22221

11211

baa

baa

baa

According to Cramers Rule:

1x = A

A1, 2x = A

A 2, and 3x =

A

A 3,

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Example 11:

Solve for x, y and z

5x - y + 7z = 4

6x - 2y + 9z= 5

2x + 8y 4z= 8

Solution:

Writing in a matrix form:

482

926

715

=

8

5

4

z

y

x

Let A =

482

926

715

.

A1 =

488

925

714

A2 =

482956

745

and A3 =

882

526

415

A =2 1A = 44 2A = -26 and 3A = -34

x =A

A1=

2

44= 22

y =AA 2 =

226 = 13

z =A

A 3=

2

34= -17

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ACTIVITY 4i

1. Solve the following system of linear equations using the inverse method.

a) 2x + y + z = 8

5x 3y + 2z = 3

7x + y + 3z = 20

b) 3x + 2y + 4z = 3

x + y + z = 2

2x y + 3z = -3

2. Solve the following system of linear equations using the Cramers Rule.

a) X1 + 2x2 X3 = 4

3X1 4X2 2X3 = 2

5X1 + 3X2 + 5X3 = -1

b) 4a 5b + 6c = 3

8a 7b 3c = 9

7a 8b + 9c = 6

3. Solve the following system of linear equations using the inverse method and Cramers Rule.

a) x + y + 2z = 1

2x + 3y + 6z = 1

3x + 2y 4z = 2

b) 3x + 2y z = 10

7x y + 6z = 8

3x + 2z 5 = 0

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