Row Echelon Calculator

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Row Echelon Calculator

When you face a system of linear equations, the fastest route to understanding it runs through the row echelon form of its augmented matrix. Reducing to a staircase reveals whether the system has no solution, one solution, or infinitely many — plus rank and free variables.

The Row Echelon Calculator on this page reduces any 3×4 augmented matrix (three equations, three unknowns) to REF via Gaussian elimination. It reports rank, consistency, free variables, and solution type.

What Is Row Echelon Form?

A matrix is in REF when: every nonzero row’s leading entry sits strictly right of the one above (staircase); all entries below each pivot are zero; zero rows sit at the bottom. Gaussian elimination produces it: per column, find a nonzero entry, swap into position, scale, clear below. A row like [0, 0, 0 | c] (c nonzero) signals an inconsistent system.

Why REF Matters

REF answers three questions in order: existence (no contradictory row → at least one solution), uniqueness (full rank → unique; rank shortfall → free variables → infinite), then values via back-substitution. In engineering, inconsistency signals impossible constraints; in economics, rank tells whether prices are uniquely determined.

How to Use the Calculator

Step 1: Form the augmented matrix (constants as 4th column).

Step 2: Type 3 rows, 4 numbers each, one row per line.

Step 3: Click Calculate.

Step 4: Read REF, Rank, System (Consistent/Inconsistent), Free Variables, Solution Type.

Worked Example 1: Unique Solution

System: x+y+z=6, 2x−y+z=3, x+2y−z=2. REF: [1,1,1|6] [0,1,0.333|3] [0,0,1|3]. Rank 3, Consistent, 0 free variables, Unique solution: z=3, y=2, x=1.

Worked Example 2: Inconsistent

System: x+y=3, 2x+2y=7, z=5. REF: [1,1,0|3] [0,0,1|5] [0,0,0|1]. Third row reads 0=1. Rank 2, Inconsistent, No solution.

Free Variables and Solution Types

Consistent + rank < variables → free variables (non-pivot columns), each adding a degree of freedom. One free variable → line of solutions; two → plane. Classification: no solution (inconsistent), unique (consistent, full rank), infinite (consistent, free variables).

Tips

  1. Build the augmented matrix first — include constants.
  2. Keep variable order consistent (x, y, z).
  3. Read the last row first — contradiction settles it.
  4. Count pivots for rank.
  5. Free variables = non-pivot columns.
  6. Verify unique solutions by substitution.

Frequently Asked Questions

1. What is an augmented matrix? Coefficients plus constants as an extra column.

2. How does REF reveal solutions? A [0,0,0|c] row (c≠0) means no solution; otherwise rank vs variables decides uniqueness.

3. What is back-substitution? Solving from the bottom pivot upward.

4. What are free variables? Non-pivot-column variables; each yields infinite solutions.

5. Can 3 equations in 3 unknowns have exactly 2 solutions? No — none, one, or infinite.

CONCLUSION

The calculator reduces any 3×4 augmented matrix to REF and classifies the system: rank, consistency, free variables, solution type. Read the staircase: contradiction rules out solutions; pivots vs variables decide uniqueness vs infinity.