Part Three

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Part Three

• Linear Algebraic Equations

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Motivation
The system of linear equations occur very
frequently in engineering applications. For small
numbers of equations (n lt 4) equations can be
solved manually by some simple techniques.
However, for four or more equations, solutions
become arduous and computers must be utilized.
Historically, the inability to solve all but
the smallest sets of equations by hand has
limited the scope of problems addressed in many
engineering applications.
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An Example of Linear Equations in Engineering
xi mass in reactor i ai properties and
characteristics of the
system bi the forcing functions acting on
the system, e.g., feed rate.
Lumped variable system
Distributed variable system
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Mathematical Background Matrix Notation
Row vector B b1 b2 . . . bm
Column vector C
A is a square matrix if n m
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Special Types of Square Matrices
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Mathematical Background Matrix Operating Rules
Matrix addition C A B (A and B must
have the same number of rows n and the same
number of columns m) cij aij bij A
B B A
Matrix multiplication C AB
(AB)C A(BC) (A(B C)
AB AC or (A B)C AC
BC AB ? BA
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Mathematical Background Matrix Operating Rules

• Inverse Matrix AA-1 A-1A I
• a non-square matrix cant have an inverse
• not every square matrix has an inverse

Transpose of a matrix B AT bij cji
Trace of a matrix tr A
Augmentation of a matrix addition of column(s)
to the original matrix Example
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Linear Algebraic Equations in Matrix Form
AX B
XT
BT
A-1AX A-1B X A-1B
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Overall Structure