Gaussian elimination

The Gaussian elimination is a generalization of the elimination method. The target is, by using the elementary operations, to get the system into a row echelon form: This kind of system lets us find a solution in a simple way.

We can check that every system is equivalent to a system in a row echelon form.

Let r the total of equations and r’ the total of non absurd equations of the system in the row echelon form, then:

–If r ≠ r’ it is an inconsistent system

–If r = r’ it is a consistent system
· If r = r’ = n it is an independent system
· If r = r’ ≠ n it is a dependent system

Where n is the number of unknowns.

NOTE: try to reorder the systems to do a11= ±1

Examples:    To discuss a system is to determine if it is a consistent or inconsistent system. If it has some parameters, we have to discuss the system depending on the parameters.

Examples:   Exercises: discuss and solve the following systems:  Solutions:
1) independent system:x = -1, y = 1, z = 3/2, t = -1/2

2) m ≠ 4 independent system: x = 1, y = 2, z = 0
m = 4 dependent system: x = 1 - λ, y = 2 - λ, z = λ; λ€R

3) independent system: x = -1, y = 3

4) b ≠ 2 independent system: x = y = ab/(2b-4), z = -2a/(b-2)
b = 2 and a = 0 dependent system: x = y = λ, z = -2λ; λ€R
b = 2 and a ≠ 0 inconsistent system