Engineering Maths 3(Week1)

KNF2033 ENGINEERING KNF2033 ENGINEERING MATHEMATICS III MATHEMATICS III

WEEK 1 & 2: •System of Linear Algebraic Equations

•Matrix

Course Synopsis

Course Approach and Assessment

Course References

Matrix - Definition

Matrix - Definition

Matrix - Definition

Matrix - Definition

Example of Matrix Usage-Solving System of Linear Equations

44.067.001.0

5.03.01.09.115.0

152.03.0

}{}{

44.05.03.01.067.09.15.0

01.052.03.0

3

2

1

321

321

321

xxx

CXA

xxxxxxxxx

Example of Matrix Usage-Solving Forces and Reactions

30

90

60

Consider a problem in structural engineering

Find the forces and reactions associated with a statically determinant truss

hinge: transmits bothvertical and horizontalforces at the surface

roller: transmitsvertical forces

Example of Matrix Usage-Solving Forces and Reactions

Example of Matrix Usage-Solving Forces and Reactions

Example of Matrix Usage-Solving Forces and Reactions

Example of Matrix-De Saint Venant-Exner Equation for Flush Wave

Singular and non-singular Matrices

Singular and non-singular Matrices

Rank of a Matrix

Rank of a Matrix

Matrix Addition and Subtraction

Matrix Scalar Multiplication

Matrix Multiplication

Matrix Multiplication

Matrix Multiplication

Matrix Transposition

Special Matrices

Square Matrix

Square Matrix

Diagonal Matrix

Unit Matrix

Unit Matrix

Null Matrix

Determinants

Determinants

Determinants

Determinants

Inverse of a Square Matrix

Step 1: Find det (A)Step 2: Find Cofactors (C) of given

matrixStep 3: Find adjoint A or CT.Step 4: Apply the formula below:

AadjA

A .)det(

11

Inverse of a Square Matrix

Example:Find the inverse of

Step 1: Find the det (A)

421134432

A

54540204520)38(4)116(3)212(2 A

Inverse of a Square Matrix

Step 2: Find cofactors (C) of given matrix

Step 3: Find adjoint of A

63432

,141442

,91343

,12132

,44142

,44243

,52134

,154114

,104213

333231

232221

131211

MMM

MMM

MMM

61514415

9410

6149144

51510 int

)1( T

ijji

ij

AofAdjo

MACofactors

Inverse of a Square Matrix

Step 4: Apply the formula

615144159410

51

61514415

9410

5-1A 1

1

1

A

adjA

A

Inverse of a Square Matrix

Now you try it!!Find the inverse of the matrix A by

adjoint matrix:

1)

442331

311A

Elementary Row Operation

Elementary row transformations on a matrix:

1.Rij: Interchange of the ith and jth rows.

2.Ri(k):Multiplication of every element of ith row by a non-zero scalar k.

3.Rij(k):Addition to the elements of ith row, of k times the corresponding elements of the jth row.

Determination of Rank of Matrix

Let A be a rectangular matrix of order m x n, apply only elementary row operations on A. Then the number of non-zero rows is the rank of A.

Example:Find the rank of

Solution:

10132230451

A

2 isA ofRank

000230451

~

230230451

~

)1(32

)2(31

R

R

Determination of Rank of Matrix

Now you try it!!Find the rank for

359674821

A