Topic 2 MATRICES & systems of linear equations

TOPIC 2MATRICES &

SYSTEMS OF LINEAR EQUATIONS

LEARNING OUTCOMESAt the end of this topic, student should be able to : Defination 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 Find minor and cofactor of 2 x 2 and 3 x 3 matrix Define Inverse Matrix Find the Inverse Matrix by using Adjoint Matrix Write a system of linear equations Solve the system of linear equations by using Inverse Matrix and

Cramer’s Rule

Definition of Matrix

A matrix is an ordered rectangular array of numbers or functions.

The numbers or functions are called elements or the entries of the matrix.

Example of a matrix

A = Order of a matrix = row x column = (m x n)

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

987654321

741

31

21

11

aaa

852

32

22

12

aaa

963

33

23

13

aaa

Types of Matrices

1. Rectangular Matrix A = 2 x

4

2. Column Matrix A = 2 x 1

87654321

21

3. Row Matrix A = 1 x 3

4. Square Matrix A = 2 x 2

5. Diagonal Matrix A = 3 x 3

321

3640

700050001

6. Zero / Null Matrix A= 2 x 2

7. Scalar Matrix A = 3 x 3

8. Upper Triangular Matrix A = 3 x 3

0000

200020002

600540321

9. Lower Triangular Matrix A = 3 x 3

10. Identity Matrix A = 3 x 3

OR A = 2 x 2

654032001

100010001

1001

Equality of Matrices

Two matrices are equal if they have the same order and same entries.

Exercises1. Find the value of x and y for the following :

(a)

(b)

(c)

(d)

4350

35y

x

10243

523yxyx

yx

3221

921

zy

yxx 4

322

Operations on MatricesAdditions / SubtractionsAdditions or subtractions of matrices can be done if they have the same dimensions whereby the two matrices must have the same number of rows and the same number of columns.

When two matrices are added or subtracted then the order of matrix should be the same.

Example :

A + B = C

121071

8750

4321

MultiplicationScalar MultiplicationExample : A =

2A = =

4771

4771

2

814

142

Multiplication of Two MatricesNecessary condition for matrix multiplication Column of first matrix should be equal to the row of

the second of matrix.

Example :

0021

0432

0024001403220312

8442

Exercises1. A = B = C =

Find : (a) A + C (b) C – A (c) 3A – 2C (d) A + 2C (e) AB

201

043

431510216

123512

Transpose of a Matrix

A matrix obtain by interchanging the rows and and columns of the original matrix. It is denoted by or .

If is an is an matrix, that is the matrix.

TA 'A

A nm TAmn

Example :

A =

0321

0231TA

Determinant of MatrixThe determinant of matrix is a unique real number 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

aaaa

A

21122211 aaaaA

A

Example :Find the value of the determinant for matrix A.

Solution :

5973

A

)97()53( A

48

Determinant for Matrix 3 x 3Let us consider a 3 x 3 matrix :

333231

232221

131211

aaaaaaaaa

A

3231

222113

3331

232112

3332

232211 aa

aaa

aaaa

aaaaa

aA

Example :

Given , find or determinant of A.

Solution :

=

512214

321A A

1214

35224

)2(5121

1

A

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

35

Minor of 2 x 2 Matrix

Let us consider matrix 2 x 2,

2221

1211

aaaa

A

2112

2211

aMaM

1234

4321

22

21

12

11

MMMM

A

ijij MM

Minor of 3 x 3 Matrix

Let us consider matrix 3 x 3,

333231

232221

131211

aaaaaaaaa

A

3332

232211 aa

aaM

32233322 aaaa

2321

131132 aa

aaM

21132311 aaaa

Example :

Find and .

Solution:

(a)

(b)

416342321

A

133164134

11 M

0444221

33 M

11M 33M

Cofactor

Cofactor for 2 x 2 Matrix

Let us consider matrix 2 x2

ijji

ij MC )1(

2221

1211

aaaa

A

....)1( 222

11 aC....)1( 21

312 aC

....)1( 123

21 aC

....)1( 114

22 aC

Example :

Given , find the cofactor for A.

Solution :

41028

A

4)4()1( 211 C

10)10()1( 312 C

2)2()1( 321 C

8)8()1( 422 C

82104

ijC

Cofactor for 3 x 3 Matrix

Let us consider matrix 3 x 3 ,

=

333231

232221

131211

aaaaaaaaa

A

3332

2322211 )1(

aaaa

C

32233322 aaaa

Example :

Given , find cofactor for A.

Solution :

3365382104

A

241593353

)1( 211

C

6)3024(3658

)1( 312

C

4218243638

)1( 413

C

924564802442624

ijC

Inverse Matrix by using Adjoint Matrix

where Steps :1. Find 2. Identify3. Identify 4. Substitute and Adj A in Inverse Matrix formulae.

AdjAA

A 11

tijCAdjA

A

ijC tijCA

Example 1 :Find if .

Solution: Step 1 :

Step 2 :

Step 3 :

Step 4 :

1A

4321

A

264 A

1234

ijC

1324t

ijC

1324

211A

21

23

12

Example 2 :Find if .

Solution: Step 1 :

Step 2 :

Step 3 :

Step 4 :

1B

125323214

B

2523

215

331

1232

4

B

34

11373145

16184

ijC

1131631418

754t

ijC

1131631418

754

3011A

3011

101

158

101

157

53

307

61

152

Systems of Linear EquationsA system of linear equations is a collection of two @ more linear equations, each containing one or more variables.

The following is a system of three equations containing three variables.

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

This is called the augmented matrix of the system.

532216234

1

zyxzyx

zyx

516

1

322234111

Exercise

Write the augmented matrix of each system.

(a)

(b)

532643

yxyx

08201

02

yxzx

zyx

Solving a system using an Inverse MatrixConsider the pair of simultenous equations

Let the matrix of coefficient be , that is

In matrix form of system above can be written as

qdycxpbyax

A

dcba

A

BAX

BAX1

ExampleSolve the following equations by using Inverse Matrix.

Solution

Step 1: Step 5 : Step 2 :

Step 3 :Step 4 :

666723

yxyx

6

76623yx

A X B

30)12(18 A

BAX 1

3266

ijC

3626t

ijCAdjA

3626

3011A

67

101

51

151

51

yx

21

yx

21

yx

(b) Solving a system using Cramer’s RuleConsider the pair of simultenous equations

Let the matrix of coefficient be , that is

Therefore by using Cramer’s Rule

for 2 x 2 Matrix

Aqdycxpbyax

dcba

A

Adqbp

X

Aqcpa

Y

Example :

Solve the system by using Cramer’s Rule 8x+5y=2 2x-4y=-10

42410

52

x 4210228

y

142

4242

)50(842

41052

x 24284

42480

4210228

y