Lecture Notes of Solution of System of Linear Algebraic Equations Using Matriix Inverse Method II

University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)






Lecture Note
Solution of System of Linear Algebraic Equations

Undergraduate Level, 3th Stage



Mr. Waleed Ali Tameemi
College of Engineering/ Babylon University
M.Sc. Civil Engineering/ the University of Kansas/ USA



2016-2017
Lecture Outline
Introduction
Matrix Inverse Method
Gauss Elimination Method
Gauss-Jordan Elimination Method
Summary
















1.0 – Introduction
For the simplicity, a system of linear algebraic equation can be solved numerically. Four different methods of solution will discuss in the following section.

2.0 – Matrix Inverse Method
This method is used in solving systems of linear algebraic equations. The following steps are required:
If the following equations are given and we asked to solve X,Y, and Z.
aX+bY+cZ=d (1)
eX+fY+gZ=h (2)
jX+kY+mZ=n (3)

Step 1: Transfer the linear equations to matrix form:
[?(a&b&c@e&f&g@j&k&m)]×[?(X@Y@Z)]=[?(d@h@n)]
[?(X@Y@Z)]=[?(a&b&c@e&f&g@j&k&m)]^(-1)×[?(d@h@n)]

[X]=[A]^(-1)×[B]
Where:
[A]: the constants that are associated with the variables [?(a&b&c@e&f&g@j&k&m)],
[X]: the variables victor [?(X@Y@Z)],
[B]: the constant vector[?(d@h@n)].
Step 2: Find the Inverse of the constant matrix [A] as follows:
Calculate the determinate of the matrix ? det(A),
Calculate the adjugate of the matrix ? adj(A),
Calculate the transpose of the adjugate of the matrix ? adj(A)^T,
Calculate the matrix inverse using the inverse equation ? ?[A]?^(-1)=(adj(A)^T)/(det?(A)).

Step 3: Find the value of the variable vector as follows:
[X]=?[A]?^(-1)×[B]