1. Chapter 6 Direct Methods for Solving Linear Systems

2. If A is nonsingular (or invertible, detA≠0)
the linear system has a unique solution.
In this chapter,direct techniques are considered to the linear system
( 2.1 )
denoted by Ax=b，where

3. 6.1 Linear systems of equations
Example 1
We first use equation E1 to eliminate the unknown x1 from E2,E3 by performing (E2-2E1)->(E2), (E2-1/2E1)->(E2). The resulting system is E1 E2 E3

4. E1 E2 E3
We first use equation E2 to eliminate the unknown x2 fromE3 by performing (E2-5/8E1)->(E2). The resulting system is E1 E2 E3

5. (2)
(1)
The system of equations (2) is now triangular (or reduced) form and can be solved for the unknowns by a backward-substitution process.

6. Gaussian elimination with backward substitution
Repeating the operations involved in Example 1 with the matrix notation results in first considering the augmented matrix:
Performing the operations as described in this Example produces the
matrices

7. Gaussian elimination with backward substitution
The general Gaussian elimination procedure applied to the linear system
is handled in a similar manner. First form the augmented matrix

8. Set
Step 1 Provided , set , and the operations corresponding to (Ei+( )E1)--->(Ei) are performed for each i=2,…,n to eliminate the coefficient of in each of these rows.
Then, we can obtain ,that is,
where

9. Step k (k=2,…,n-1)

10. k=1,…,n-1
At last, we can get a linear system which is triangular, thus, backward substitution can be performed.