RREF Calculator (Row Echelon Form)
Follow matrix reduction in MiniWebtool’s RREF Calculator (Row Echelon Form). Each pivot, row swap, and elimination leads to reduced row echelon form, with rank, nullity, and solution information for an augmented matrix.
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Enter a matrix up to 10×10, or an augmented one, to get its reduced row echelon form with every row operation, the pivots, rank and solution type.
About RREF Calculator (Row Echelon Form)
The RREF Calculator (Reduced Row Echelon Form Calculator) performs Gauss-Jordan elimination on any matrix, showing every row operation along the way. Whether you are solving a system of linear equations, finding the rank of a matrix, or identifying pivot and free columns, this tool gives you the complete step-by-step solution with exact fractional arithmetic — no rounding errors.
At a Glance
RREF Calculator (Row Echelon Form) reduces a matrix to reduced row echelon form by Gauss-Jordan elimination. It shows every row operation, plus rank, nullity, pivot columns and free variables, using exact fractions.
| Feature | What you get | Good for |
|---|---|---|
| Matrix grid | Set the number of rows and columns with the plus and minus controls, then enter integers, decimals or fractions such as 1/3. | Entering coefficient matrices of different sizes |
| Exact fractions | Results use exact fractional arithmetic, so no rounding errors build up during elimination. | Checking homework where answers must be exact |
| Step navigator | Move through each row swap, scaling and addition with the arrow buttons, or press Play to animate the whole reduction. | Learning how Gauss-Jordan elimination proceeds |
| Augmented mode | Tick Augmented [A|b] to treat the last column as constants. The result says whether the system has a unique solution, infinitely many, or none. | Solving systems of linear equations |
| Rank and nullity | Reports rank, nullity, pivot columns and free variables, and the rank plus nullity equals the number of columns. | Linear algebra courses covering column space and null space |
| Example presets | Buttons load a 3×3 basic case, unique solution, infinite solutions, inconsistent system, tridiagonal matrix or 2×2 matrix. | Seeing each type of outcome without typing a matrix |
What Is Reduced Row Echelon Form (RREF)?
A matrix is in Reduced Row Echelon Form when it satisfies all of the following conditions:
How to Use the RREF Calculator
Step 1. Set the number of rows and columns using the +/− controls.
Step 2. Enter your matrix values into the grid cells. You can enter integers, decimals, or fractions like 1/3. Use Tab, Enter, or arrow keys to navigate between cells.
Step 3. If you are solving a system of equations, check Augmented [A|b] to mark the last column as the constants vector.
Step 4. Click Calculate RREF.
Step 5. Review the results: the RREF matrix, rank, nullity, pivot columns, and free variables. Use the step navigator or the Play button to watch each row operation unfold.
Row Echelon Form vs. Reduced Row Echelon Form
| Property | REF (Row Echelon Form) | RREF (Reduced Row Echelon Form) |
|---|---|---|
| Zeros below pivots | Yes | Yes |
| Zeros above pivots | Not required | Yes |
| Pivots equal to 1 | Not required | Yes |
| Uniqueness | Not unique | Unique for every matrix |
| Method | Forward elimination | Forward + back elimination |
Understanding the Results
Rank is the number of pivot positions, representing the dimension of the column space (or row space). Nullity is the number of non-pivot columns, representing the dimension of the null space. The Rank-Nullity Theorem guarantees: Rank + Nullity = number of columns.
For augmented matrices \([A|b]\), the solution type depends on the RREF:
| Condition | Solution Type |
|---|---|
| Row \([0\; 0\; \cdots\; 0\; |\; c]\) with \(c \neq 0\) | No solution (inconsistent) |
| Every column of A is a pivot column | Unique solution |
| Some non-pivot columns in A, no inconsistent rows | Infinitely many solutions |
Elementary Row Operations
The three operations used to compute RREF preserve the solution set of a linear system:
Frequently Asked Questions
What is Reduced Row Echelon Form (RREF)?
RREF is the canonical form of a matrix obtained through Gauss-Jordan elimination. In RREF, each leading entry (pivot) is 1, each pivot is the only nonzero entry in its column, and pivot positions move strictly to the right and down. Every matrix has a unique RREF.
What is the difference between REF and RREF?
Row Echelon Form (REF) only requires zeros below each pivot, while Reduced Row Echelon Form (RREF) additionally requires zeros above each pivot and all pivots equal to 1. RREF is unique for any given matrix, but REF is not.
How do you find the rank of a matrix using RREF?
The rank of a matrix equals the number of pivots (leading 1s) in its RREF. Pivot columns are the columns containing these leading 1s. The nullity equals the number of columns minus the rank, which is also the number of free variables.
How do you solve a system of equations using RREF?
Write the augmented matrix [A|b] for the system Ax = b, then reduce it to RREF. If any row has the form [0 0 ... 0 | c] with c nonzero, the system is inconsistent (no solution). Otherwise, pivot columns give determined variables and non-pivot columns correspond to free variables that can take any value.
What row operations are used to find RREF?
Three elementary row operations are used: (1) swapping two rows, (2) multiplying a row by a nonzero scalar, and (3) adding a multiple of one row to another. These operations do not change the row space or solution set of a linear system.
Reference this content, page, or tool as:
"RREF Calculator (Row Echelon Form)" at https://MiniWebtool.com/rref-calculator-row-echelon-form/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-09
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