engineering maths 3(week1)
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Lecture notesTRANSCRIPT
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KNF2033 ENGINEERING KNF2033 ENGINEERING MATHEMATICS III MATHEMATICS III
WEEK 1 & 2: •System of Linear Algebraic Equations
•Matrix
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Course Synopsis
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Course Approach and Assessment
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Course References
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Matrix - Definition
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Matrix - Definition
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Matrix - Definition
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Matrix - Definition
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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
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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
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Example of Matrix Usage-Solving Forces and Reactions
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Example of Matrix Usage-Solving Forces and Reactions
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Example of Matrix Usage-Solving Forces and Reactions
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Example of Matrix-De Saint Venant-Exner Equation for Flush Wave
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Singular and non-singular Matrices
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Singular and non-singular Matrices
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Rank of a Matrix
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Rank of a Matrix
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Matrix Addition and Subtraction
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Matrix Scalar Multiplication
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Matrix Multiplication
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Matrix Multiplication
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Matrix Multiplication
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Matrix Transposition
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Special Matrices
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Square Matrix
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Square Matrix
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Diagonal Matrix
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Unit Matrix
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Unit Matrix
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Null Matrix
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Determinants
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Determinants
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Determinants
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Determinants
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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
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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
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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
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Inverse of a Square Matrix
Step 4: Apply the formula
615144159410
51
61514415
9410
5-1A 1
1
1
A
adjA
A
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Inverse of a Square Matrix
Now you try it!!Find the inverse of the matrix A by
adjoint matrix:
1)
442331
311A
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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.
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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
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Determination of Rank of Matrix
Now you try it!!Find the rank for
359674821
A
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