Enter Your Matrix
Use integers, decimals, or fractions like 3/4.
Example Data Table
This example shows a 3 × 4 augmented matrix and its reduced form.
| Case | Input Matrix | RREF Output | Interpretation |
|---|---|---|---|
| Sample system | [1, 2, -1 | 8] [2, -1, 3 | 9] [-1, 1, 2 | 3] |
[1, 0, 0 | 3.5] [0, 1, 0 | 3.1] [0, 0, 1 | 1.7] |
Unique solution with three pivots and zero free variables. |
| Dependent system | [1, 2, 1 | 4] [2, 4, 2 | 8] [0, 0, 1 | 3] |
[1, 2, 0 | 1] [0, 0, 1 | 3] [0, 0, 0 | 0] |
Infinite solutions because one variable remains free. |
Formula Used
The solver applies Gauss-Jordan elimination. It repeatedly chooses a pivot, scales the pivot row, and clears every other entry in that pivot column.
R_i ← R_i / a_{ij} makes the pivot equal to 1.
R_k ← R_k - a_{kj}R_i removes nonzero entries above and below the pivot.
For augmented systems, the solver also checks:
- Coefficient rank versus augmented rank to test consistency.
- Nullity = number of variables − coefficient rank.
- Determinant when the coefficient matrix is square.
Once every pivot column contains one leading 1 and all other entries in that column are 0, the matrix is in reduced row echelon form.
How to Use This Calculator
- Choose the number of rows and columns for your matrix.
- Enable the constants option when the last column represents equation totals.
- Enter matrix values using whole numbers, decimals, or fractions.
- Adjust precision and tolerance when tiny rounding differences matter.
- Click Solve to RREF to compute the reduced matrix.
- Review rank, pivots, nullity, determinant, and solution classification.
- Use CSV or PDF export buttons to save the result.
FAQs
1. What is reduced row echelon form?
It is a matrix form where each pivot is 1, every pivot column has zeros elsewhere, and pivots move rightward as rows go down. This layout makes solution structure easy to read.
2. Can I enter fractions instead of decimals?
Yes. You can enter values like 1/2, -3/4, or 7/5. The solver converts them to decimal values internally, then displays results using your selected precision.
3. What does the constants option do?
It treats the last column as the right-hand side of a linear system. That lets the solver classify the system as unique, inconsistent, or infinitely solvable.
4. Why does tolerance matter?
Tolerance decides when a very small number should count as zero. This is useful for decimal-heavy matrices where arithmetic can leave tiny rounding leftovers.
5. What does nullity tell me?
Nullity measures how many independent free directions remain after reduction. For systems, it equals the number of free variables and indicates whether infinitely many solutions exist.
6. When is the determinant shown?
The determinant appears only when the coefficient matrix is square. A zero determinant means the square matrix is singular and does not have full independent rows or columns.
7. Can this calculator solve non-square matrices?
Yes. Reduced row echelon form works for rectangular matrices too. The solver still reports pivots, rank, nullity, and free columns even when rows and columns differ.
8. What do the export buttons save?
They save your summary metrics, original matrix, reduced matrix, and any solution interpretation. This is useful for assignments, notes, checking work, or sharing results.