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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
