Row Echelon Form Calculator
Reduce a two- or three-equation linear system to REF or RREF, identify pivots and rank, classify the solution set, and export the complete calculation to a real Excel workbook.
Enter the augmented matrix
Accepted entries: integers, decimals using a period, or fractions such as -3/4. Commas and scientific notation are rejected to prevent ambiguous interpretation.
Live result
Solution classification
Unique solution
x = 3, y = 14, z = 15
Coefficient rank
3
Augmented rank
3
Pivot columns
x, y, z
Row operations
7
Reduced row echelon matrix
Solution set
Elimination steps
| Step | Row operation | Matrix after the operation |
|---|
The table records the calculator's deterministic pivot choices. Different valid row-operation sequences can produce different REF matrices, while a matrix's true RREF is unique.
How to use the row echelon form calculator
What this calculator does
This calculator applies elementary row operations to the augmented matrix of a two-variable or three-variable linear system. It can stop at row echelon form (REF), which is suited to back-substitution, or continue to reduced row echelon form (RREF), where every pivot is 1 and every pivot column is cleared above and below that pivot. It also reports the coefficient rank, augmented rank, pivot columns, row-operation count, and the resulting solution set. It does not solve nonlinear equations, inequalities, symbolic expressions, or systems with more than three variables.
When to use it
Use it to check hand calculations in an algebra or linear algebra course, convert a system into an easier matrix form, determine whether a system has one solution, no solution, or infinitely many solutions, and inspect which variables are pivot variables versus free variables. The method follows the same augmented-matrix workflow described in OpenStax's Gaussian elimination lesson.
How to calculate
- The calculator opens with a complete demonstration system and a validated Excel workbook ready to download. Choose Number of equations: either 2 equations / 2 variables or 3 equations / 3 variables.
- Choose Output form. Select RREF for a fully reduced result that exposes the solution directly, or REF when you want the forward-elimination form used before back-substitution.
- Replace the demonstration coefficients and constants in the augmented matrix. Every active box is required. Use integers, period-decimals, or fractions such as 5/2. Enter 0 when a variable is absent from an equation.
- Read the live classification, rank values, pivot columns, output matrix, solution set, and elimination table. Use Download Excel to export the current validated state. Reset clears the demonstration data, removes results, and disables Excel export until a complete valid system is entered again.
Input guide
Number of equations is a required select control. Choose 2 or 3; it also sets the number of variables. Changing from 3 to 2 temporarily excludes the third equation and every z coefficient from the calculation. A common mistake is to expect a rectangular matrix; this interface intentionally mirrors the reference workflow's square systems.
Output form is required. RREF performs Gauss-Jordan elimination, while REF performs forward Gaussian elimination only. The choice changes the output matrix and operation count, but it does not change the underlying solution classification.
The first row consists of Equation 1, x coefficient, Equation 1, y coefficient, the optional Equation 1, z coefficient, and Equation 1, constant. The second row has the corresponding four Equation 2 entries. Three-equation mode adds Equation 3, x coefficient, Equation 3, y coefficient, Equation 3, z coefficient, and Equation 3, constant. All active entries are required real numbers. For example, x + y + z = 32 is entered as 1, 1, 1, 32. Higher or lower coefficients alter pivot selection and elimination factors; changing a constant changes the right-hand side without changing the coefficient matrix. Do not leave a missing variable blank – enter 0. Commas, mixed numbers, scientific notation, division by zero, and symbols such as x are rejected rather than silently reinterpreted.
Output guide
Solution classification is “Unique solution,” “Infinitely many solutions,” or “No solution.” It is determined by comparing the Coefficient rank with the Augmented rank and the number of variables. Equal ranks matching the variable count mean a unique solution. Equal ranks below the variable count mean free variables and infinitely many solutions. A larger augmented rank means the equations are inconsistent.
Pivot columns lists the variables whose columns contain leading entries in RREF. Row operations counts swaps, row scaling, and row replacement steps after the initial matrix. Reduced row echelon matrix or Row echelon matrix is the selected transformed augmented matrix. Solution set gives exact decimal-style values for a unique solution, a parametric description for an infinite family, or an inconsistency statement. The Elimination steps table contains the step number, the operation performed, and a snapshot of the matrix after that operation. These are computed results, not numerical approximations of a physical measurement; very small floating-point residue is normalized to zero.
Worked example
The startup example represents x + y + z = 32, -x + 2y = 25, and -y + 2z = 16. Its augmented matrix is [1, 1, 1 | 32], [-1, 2, 0 | 25], and [0, -1, 2 | 16]. Forward elimination first adds row 1 to row 2, giving [0, 3, 1 | 57], then adds one third of row 2 to row 3, giving [0, 0, 7/3 | 35]. Continuing to RREF produces the identity coefficient matrix with constants 3, 14, and 15. Therefore the first-open output is Unique solution, coefficient rank 3, augmented rank 3, pivot columns x, y, z, and solution x = 3, y = 14, z = 15.
How row reduction preserves solutions
Swapping two rows, multiplying a row by a nonzero scalar, or adding a multiple of one row to another creates a row-equivalent matrix. These operations preserve the solution set because each operation is reversible. For a more formal treatment of echelon shape and the elimination algorithm, see the MIT notes on Gaussian elimination.
REF versus RREF
REF requires zero rows at the bottom, pivots moving to the right as you move downward, and zeros below every pivot. RREF adds two stricter conditions: each pivot equals 1 and is the only nonzero entry in its column. REF is not unique because many row-equivalent staircase matrices are possible. RREF is unique, making it especially useful for comparing answers and identifying free variables. The LibreTexts RREF algorithm provides additional examples of the complete reduction process.