Direct Methods for Solving Linear Systems
Linear systems of equations are associated with many problems in engineering and science, as well as with applications of mathematics to the social sciences and the quantitative study of business and economic problems.
In this chapter we consider direct methods for solving a linear system of n equations in n variables. Such a system has the form
In this system we are given the constants ai j, for each i, j = 1, 2, . . . , n, and bi, for each i = 1, 2, . . . , n, and we need to determine the unknowns x1, . . . , xn.
The system can be presented by matrices as follow:
or
The above matrix is called the Augmented Matrix.
Example2:
Use Gaussian Elimination Method to find the solution of the linear system
E1 : x1 ? x2 + 2x3 ? x4 = ?8,
E2: 2x1 ? 2x2 + 3x3 ? 3x4 = ?20,
E3 : x1 + x2 + x3 = ?2,
E4 : x1 ? x2 + 4x3 + 3x4 = 4,
Solution:
1- Forward Elemination:
We eliminate x1 from E2, E3, and E4 as follow
(E2 ? 2E1) ? (E2), (E3 ? E1) ? (E3), and (E4 ? E1) ? (E4),
We get
E1 : x1 ? x2 +2x3? x4= ?8,
E2: - x3 ? x4 = ?4,
E3 : 2x2 - x3 + x4 = 6,
E4 : 2x3 + 4x4 = 12,
In E2, note that the coefficient of x2 is zero, so we replace E2 and E3 by each other
E1 : x1 ? x2 +2x3?x4= ?8,
E2 : 2x2 - x3 + x4 = 6,
E3: - x3 ? x4 = ?4,
E4 : 2x3 + 4x4 = 12,
Now, we need only to eliminate x3 from E4 as follow: (E4 + 2E3) ? (E4),
We get
E1 : x1 ? x2 +2x3?x4= ?8,
E2 : 2x2 - x3 + x4 = 6,
E3: - x3 ? x4 = ?4,
E4 : 2x4 = 4,
2- Backward Subtraction:
From E4, x4=2, from E3, x3= 2, from E2, x2=3, from E1, x1=7.
Exercises:
Use the Gaussian Elimination Method to solve the following linear systems,
a. 2x1 ? 1.5x2 + 3x3 = 1,
?x1 + 2x3 = 3,
4x1 ? 4.5x2 + 5x3 = 1.