Lecture Notes of Solution of System of Linear Algebraic Equations Using Gauss-JordanElimination 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
















Gauss-Jordan Elimination Method
This method is used in solving systems of linear algebraic equations. It is similar to Gauss Elimination Method and as follows:

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)]

Step 2: Rearrange the matrix rows and columns by multiplying any column or any row by a factor and add any row to any other row and any column to other column and/or replacing any row (column) by any other row (column) until making the value of the main diagonal equal to (1) and the values of all elements that located above and below the main diagonal equal to (0).
[?(1&0&0@0&1&0@0&0&1)]
Note that when you multiply a column and/or a row by a factor(s) and added to another row and/or column, the other side of the equation [?(d@h@n)]is also changed.
Result of step 2:
[?(1&0&0@0&1&0@0&0&1)]×[?(X@Y@Z)]=[?(r@t@s)]
Step 3: Solve the matrix equation and as follows:
X=r
Y=t
Z=s