In the previous section we studied systems with unique
solutions. In this section we will
study systems of linear equations that have infinitely many solutions and those
that have
no solution. We also will study systems in which the number of variables is not
equal to
the number of equations in the system .
A System of Equations with an Infinite Number of
Solutions
Example 1: The fol lowing augmented matrix in
row- reduced form is equivalent to the
augmented matrix of a certain system of linear equations. Use this result to
solve the
A System of Equations That Has No Solution
Example: Given the following system
In using the Gauss-Jordan elimination method the following
equivalent matrix was
obtained (note this matrix is not in row-reduced form, let’s see why):
Systems with No Solution
If there is a row in the augmented matrix containing all
zeros to the left of the vertical
line and a nonzero entry to the right of the line, then the system of equations
has no
solution.
Theorem
I. If the number of equations is greater than or equal to
the number of variables in a
linear system, then one of the following is true:
a. The system has no solution.
b. The system has exactly one solution.
c. The system has infinitely many solutions.
II. If there are fewer equations than variables in a
linear system, then the system either
has no solution or it has infinitely many solutions.
Example 2: Solve the system of linear equations
using the Gauss-Jordan elimination
Example 3: Solve the system of linear equations
using the Gauss-Jordan elimination
Example 4: Solve the system of linear equations
using the Gauss-Jordan elimination
Example 5: Solve the system of linear equations
using the Gauss-Jordan elimination
