Calculate reduced row echelon form (RREF) of any matrix with step-by-step Gaussian elimination. Supports 2×2 to 4×5 augmented matrices.
Enter integers, decimals, or fractions (e.g. 1/2)
Shaded column = constants (b vector)
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The Reduced Row Echelon Form (RREF) of a matrix is computed using Gaussian-Jordan elimination. Start by identifying the leftmost nonzero column, swap rows to bring a nonzero entry to the top, scale it to 1 (the pivot), then eliminate all other entries in that column. Repeat for each subsequent column. Our RREF calculator handles integer, decimal, and fraction entries automatically.
RREF is a standardized form of a matrix used in linear algebra to solve systems of equations, find matrix rank, and determine null spaces. A matrix is in RREF when: (1) each non-zero row has a leading 1 (pivot), (2) each pivot column has zeros in all other rows, (3) pivots move strictly to the right as you go down rows, and (4) all-zero rows are at the bottom.
Row Operation Types
Our calculator uses exact rational arithmetic internally, so results like 1/3 or 2/7 display precisely without floating-point rounding errors.
Every row swap, scaling, and elimination operation is shown so you can follow along or check your own work for homework or exams.
The calculator automatically identifies whether the system has a unique solution, infinitely many solutions, or no solution at all.
Toggle between Row Echelon Form (REF) and Reduced Row Echelon Form (RREF) depending on what your textbook or course requires.
RREF is the standard method for solving systems with 2, 3, or more equations. Augment the coefficient matrix with your constants column and compute RREF to read off the solution directly.
The rank of a matrix equals the number of nonzero rows in its row echelon form. This is fundamental for determining whether a system is consistent and whether solutions are unique.
Columns without a pivot (non-pivot columns) correspond to free variables. Use RREF to identify these columns and parameterize the null space of a matrix.
Place your vectors as rows (or columns) and reduce. If all rows are nonzero in the result, the vectors are linearly independent. A zero row signals dependence.
REF (Row Echelon Form) requires that each nonzero row has a leading entry (pivot) to the right of the pivot in the row above, and all-zero rows are at the bottom. RREF additionally requires each pivot to be 1 and to be the only nonzero entry in its column. RREF is unique for any given matrix; REF is not.
Yes. Enter fractions in the format 1/2, -3/4, etc. The calculator uses exact rational arithmetic throughout the elimination process, so your results will also display as fractions without rounding.
After reduction, look for a row where all coefficient entries are zero but the constant column (last column in an augmented matrix) is nonzero — for example [0 0 0 | 5]. This represents the equation 0 = 5, which is a contradiction. The system has no solution.
The rank of a matrix is the number of pivot rows (nonzero rows) in its row echelon form. It tells you the dimension of the column space and row space. For an m×n coefficient matrix of a system with n unknowns, rank = n means a unique solution, rank < n means free variables (infinite solutions), and an augmented inconsistency means no solution.
Pivot columns are the columns that contain a leading 1 (pivot) in the RREF. They correspond to the basic variables in the system. Non-pivot columns correspond to free variables. The number of pivot columns equals the rank of the matrix.