A simple example
Row echelon form
When bad things happen to good matrices
Math 160
§2.1–Systems of Linear Equations and
Elimination
Tom Lewis
Fall Semester
2013
A simple example
Row echelon form
Outline
A simple example
Row echelon form
When bad things happen to good matrices
When bad things happen to good matrices
A simple example
Row echelon form
When bad things happen to good matrices
Problem
Solve the system of equations
2x1 + 3x2 = 17
4x1 + 5x2 = 31
Discuss row operations, pivots, equivalent systems, and
back-substitution.
A simple example
Row echelon form
When bad things happen to good matrices
Equivalent systems of equations
8
4x + y = 19
8
6
6
4
4
(4, 3)
2
2x + 3y = 17
2
4
6
8
(4, 3)
2
y =3
2x + 3y = 17
2
4
6
8
A simple example
Row echelon form
When bad things happen to good matrices
Example (Augmented matrix)
The previous system of equations can be represented as an
augmented matrix:
2 3 17
4 5 31
A simple example
Row echelon form
When bad things happen to good matrices
Definition (Elementary row operations)
There are three types of elementary row operations:
1. Multiply a row by a non-zero constant.
2. Interchange two rows.
3. Add a multiple of a copy of one row to another row.
Theorem
If a system of matrices is altered by a row operation, then the
resulting system is equivalent.
A simple example
Row echelon form
When bad things happen to good matrices
Problem
Solve the system of equations corresponding to the augmented
matrix:


3
1 2 −1
 2 7 −11 −9 
1 4 −5 −3
A simple example
Row echelon form
When bad things happen to good matrices
Definition
A matrix is in row echelon form (REF) provided that
1. When all entries in a row are all zeros, this row appears below
all rows that contain a non-zero entry.
2. When two non-zero rows are compared, the first non-zero
entry, called the leading entry, in the upper row is to the left
of the leading entry in the lower row.
A simple example
Row echelon form
When bad things happen to good matrices
Problem
Each matrix below is not in REF. What is wrong? How can we fix
it?




1 0 −4 8
1 3 3 7
 0 0 0
 0 0 2 4 
0 
0 3 6 14
0 0 1 −5
A simple example
Row echelon form
When bad things happen to good matrices
Problem
Place the following matrix into REF and describe its solution set:


1 −1 −2 5
 0 2 −4 6 
0 0
0 0
A simple example
Row echelon form
When bad things happen to good matrices
Figure: The solution set of the system is a line.
A simple example
Row echelon form
When bad things happen to good matrices
Problem
Describe the solution set of a system of equations with the
following REF:


1 −1 −2 5
 0 2 −4 6 
0 −4 8 3
A simple example
Row echelon form
When bad things happen to good matrices
Definition (consistent and inconsistent systems)
1. If a system of equations has at least one solution, then it is
called consistent. A consistent system can have a unique
solution or infinitely many solutions.
2. If a system of equations has no solution, then it is called
inconsistent.