Chapter 2.3: GaussJordan Elimination
Systems with an Infinite Number of Solutions
•
Today we will look at systems where there are an infinite num
ber of solutions, or no solution at all.
•
Also, we will look at systems in which the number of equations
is not equal to the number of variables.
•
In all cases, the means of finding a solution are the same:
1. Write down the corresponding augmented matrix for the
given system.
2. Rowreduce the matrix until it is in rowreduced form.
3. Determine what the solutions are.
•
The only thing that changes is step three – determining whether
there is one solution, an infinite number of solutions, or no
solution.
Example:
Solve the following system:
x
+ 2
y
= 3
4
x
+ 8
y
= 20
First, we write down the corresponding augmented matrix:
bracketleftbigg
1
2
3
4
8
20
bracketrightbigg
Now, we rowreduce:
bracketleftbigg
1
2
3
4
8
20
bracketrightbigg
R
2

4
R
1
→
bracketleftbigg
1
2
3
0
0
8
bracketrightbigg
At this point, I claim that the matrix is in rowreduced form. Go
back to the four rules for rowreduced form; you should confirm that
this matrix satisfies all four conditions.
Now, look at the last row of the matrix.
The corresponding
equation is 0
x
+ 0
y
= 8, or 0 = 8.
This is a contradiction.
As
before, when we end up with a contradiction, we conclude that the
system has no solution.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
If there is a row of zeros in the coefficient matrix, with
a nonzero number in the augmented column, then the
given system has no solution.
Example:
Solve the given system.
x
+
y
+ 3
z
= 2
2
x
+ 3
y
+
z
= 9
3
x
+ 4
y
+ 4
z
= 11
We begin, as always by writing down the corresponding aug
mented matrix, and rowreducing until the matrix is in rowreduced
form.
1
1
3
2
2
3
1
9
3
4
4
11
R
2

2
R
1
→
1
1
3
2
0
1

5
5
3
4
4
11
R
3

3
R
1
→
1
1
3
2
0
1

5
5
0
1

5
5
R
1

R
2
→
1
0
8

3
0
1

5
5
0
1

5
5
R
3

R
2
→
1
0
8

3
0
1

5
5
0
0
0
0
Again, I claim that this matrix is in rowreduced form. At the
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 stephenlang
 Calculus, Equations, GaussJordan Elimination, Complex number, Underdetermined Systems

Click to edit the document details