Results
- Classification
- —
- Solution
- —
- Parametric form
- —
- Rank summary
- —
Elimination steps
Matrix snapshots
Snapshots show the augmented matrix after each operation; pivot cells are highlighted to match the step log.
Teacher mode
Toggle reminders about partial pivoting and keeping fractions exact before rounding for presentations.
How to use the RREF solver
Enter the coefficients of a 2×2 or 3×3 augmented matrix, choose whether you want REF or full RREF, and then solve once to see the full reduction. Use the step button when you want to reveal one row operation at a time for teaching or self-checking.
How it works
The solver performs elementary row operations on the augmented matrix [A|b]. It chooses pivots, swaps rows when needed, scales pivot rows, and eliminates the remaining entries in the pivot column. The matrix snapshots show the state after each operation so the algebra and the matrix view stay aligned.
When to use REF or RREF
Use REF when you mainly want the elimination path and a triangular form. Use RREF when you want the cleanest final classification, explicit pivot columns, and a clearer parametric description of any free variables.
Common mistakes to avoid
- Typing the wrong augmented entry in the rightmost column.
- Switching between fraction and decimal view too early and losing exact structure.
- Confusing an inconsistent row with a free-variable row.
- Reading the final rank summary without checking the elimination steps that produced it.
Worked interpretation
If the reduced form contains a row like 0 0 0 | 5, the system is inconsistent and has no solution. If one or more columns never become pivot columns while the augmented matrix stays consistent, the solver labels those variables as free and reports the parametric family of solutions.
See also
FAQ
What does the elimination log include?
Every pivot choice, row swap, row scaling, and elimination step is written in Ri notation so you can match the algebra with each matrix snapshot.
How are infinite or no solutions detected?
After reduction, the solver compares rank(A) and rank([A|b]). An inconsistent row means no solution, while a rank smaller than the number of variables means at least one free variable and therefore infinitely many solutions.
When should I use the step button?
Use the step button when you want to teach, debug, or verify one row operation at a time. It is especially useful for checking why a row swap happened or why a pivot column changed.
Reading the output correctly
Classification block
The classification field summarizes whether the system has one solution, infinitely many solutions, or no solution. Read it together with the rank summary rather than in isolation.
Solution and parametric form
A unique solution is reported directly. If free variables remain, the solver writes a parametric form so you can see how pivot variables depend on the free ones.
Teacher mode
Teacher mode adds reminders about partial pivoting and exact fractions so you can present the same reduction sequence without hiding why each row operation was chosen.
Fraction vs decimal view
Fraction view preserves exact structure and is best for instruction. Decimal view is useful for quick reading, but it can hide patterns such as exact zero rows or simple rational pivots.
Matrix snapshots
The snapshots are not decorative: they are there so you can verify that each logged row operation actually transformed the matrix you entered.