Matrix Setup
Example Data Table
| Name | Size | Description | |
|---|---|---|---|
| Augmented 3×4 system | 3 × 4 | Classic system producing a unique solution after reduction. | |
| Underdetermined | 2 × 3 | Free variable appears; final pivot set not full. | |
| Fractional entries | 3 × 3 | Rational inputs exercise exact arithmetic pipeline. |
You may enter numbers like -7, 3/5, or -2.75. All computations use exact fractions.
Results
Final RREF
Operation Log
| # | Operation | Pivot | Snapshot |
|---|
Formula Used
The calculator performs Gauss–Jordan elimination with exact rational arithmetic. At each pivot column k, we:
- Find a row r with nonzero a[r,k]; swap into position r.
- Scale row r by the reciprocal of its pivot to create a leading one.
- Eliminate the pivot column in all other rows: Rj := Rj − a[j,k] · Rr.
The result is reduced row echelon form: each leading one is the only nonzero in its column; leading ones move strictly rightward; zero rows, if any, appear at the bottom.
How to Use
- Choose matrix size, then build the grid.
- Enter values. Fractions like 7/3 or decimals like -1.25 are accepted.
- Click Compute RREF. Steps will document swaps, scales, and eliminations.
- Use Download CSV or Download PDF to export the log and final matrix.
What is Reduced Row Echelon Form?
- Every nonzero row begins with a leading one (pivot).
- Each pivot is the only nonzero in its column.
- Pivots move strictly rightward as you go down the rows.
- Any all‑zero rows appear at the bottom of the matrix.
These properties uniquely determine the RREF for a given matrix.
Interpreting RREF: Rank, Pivots, and Solution Sets
Rank equals the number of pivot columns. Free variables correspond to non‑pivot columns.
- Unique solution: Full column rank for variables; no contradictions.
- Infinitely many solutions: At least one free variable; no contradictions.
- No solution (inconsistent): A row like [0 … 0 | 1] in an augmented matrix.
For an augmented matrix [A | b], compare rank(A) and rank([A|b]) to decide consistency.
Elementary Row Operations Used
| Operation | Form | Purpose |
|---|---|---|
| Swap rows | Ri ↔ Rj | Move a nonzero pivot into the working row. |
| Scale a row | Ri := c · Ri, c ≠ 0 | Create a leading one by dividing by the pivot value. |
| Row replacement | Ri := Ri − k · Rp | Zero out other entries in the pivot column. |
These operations preserve solution sets and lead deterministically to RREF.