Gaussian Elimination With A Parameter

This example shows how to classify the solution set of a parameter-dependent linear system.

Problem:

Find the values of such that the following system has:

  1. no solutions
  2. a unique solution
  3. infinitely many solutions

System:

Solution:

1. Write the compact system.

2. Form the augmented matrix.

3. Perform Gaussian elimination.
Apply the row operation :

4. Evaluate cases.

  • Case 1: .
    The pivots are all nonzero , so the system has a unique solution.

  • Case 2: .
    After substitution we find , so there are infinitely many solutions.

  • Case 3: .
    The augmented matrix becomes

    Apply :

    Again , so there are infinitely many solutions.

Conclusion:

  • No value of produces no solutions.
  • For the system has a unique solution.
  • For or the system has infinitely many solutions.