Example of using the echelon form to determine if a linear system is consistent.

Created over 8 years ago, updated 10 days ago

Example: Using Echelon Form to Determine Consistency

Determine whether the following system is consistent:

x + 2y - z =  1
2x + 4y - 2z =  3
-x - 2y + z = -1

Step 1: Write augmented matrix

[ 1  2 -1 |  1]
[ 2  4 -2 |  3]
[-1 -2  1 | -1]

Step 2: R₂ → R₂ - 2R₁, R₃ → R₃ + R₁

[ 1  2 -1 |  1]
[ 0  0  0 |  1]
[ 0  0  0 |  0]

Step 3: Analyze echelon form

Row 2 reads [0 0 0 | 1], which corresponds to the equation 0 = 1. This is a contradiction.

Conclusion: The system is inconsistent -- it has no solution.

Contrast with a consistent system:

x + 2y - z = 1
2x + 4y - 2z = 2
-x - 2y + z = -1

Augmented matrix after reduction:

[ 1  2 -1 |  1]
[ 0  0  0 |  0]
[ 0  0  0 |  0]

No contradictory row. The system is consistent with free variables y and z, giving infinitely many solutions: x = 1 - 2y + z for any y, z in R.