RREF Calculator (Reduced Row Echelon Form)
Row-reduce any m×n matrix to its unique reduced row echelon form using Gauss-Jordan elimination with partial pivoting.
3
rankPivot columns (rank)
75%
Free columns (nullity)
25%
How does this calculator work?
Enter a matrix row by row and this tool applies Gauss-Jordan elimination with partial pivoting to return its unique reduced row echelon form. It also reports the pivot columns, the rank (number of pivots), and the nullity, so you can analyze solvability, independence, and dimension at a glance.
Formula
How this is calculated
Enter the matrix A as text with one row per line and entries separated by spaces or commas. The calculator parses it into an m×n grid and verifies that every row has the same length, otherwise the input is rejected. Each numeric entry must be finite.
Gauss-Jordan elimination then processes the columns left to right. For the current pivot position it applies partial pivoting: it scans the current column at and below the pivot row and selects the entry with the largest absolute value, swapping that row up for numerical stability. If the largest magnitude in the column is effectively zero (below a 1e-10 tolerance) the column has no pivot and is skipped. Otherwise the pivot row is normalized by dividing through by the pivot value, and multiples of it are subtracted from every other row so the pivot column holds a single 1 with zeros elsewhere.
The result is the unique reduced row echelon form. The pivot columns identify the basic variables, the rank equals the number of pivots, and the nullity (number of free columns) is the column count minus the rank. Values within the rounding tolerance are snapped to exactly 0 to remove floating-point noise and negative-zero artifacts.
Frequently asked questions
Row echelon form only requires zeros below each leading entry. Reduced row echelon form goes further: every pivot is 1 and is the only nonzero entry in its column, which makes the result unique for a given matrix.
The rank is the number of pivot columns, also shown directly. It equals the count of nonzero rows in the RREF and tells you the dimension of the row and column spaces.
Yes. Append the constants as an extra column. After reduction, pivot columns among the variable columns give the solved variables, and a pivot in the final column signals an inconsistent (no-solution) system.
Also known as
TG we-Calculate Editorial Team. (2026). RREF Calculator (Reduced Row Echelon Form) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rref-calculator
TG we-Calculate Editorial Team. "RREF Calculator (Reduced Row Echelon Form)." TG we-Calculate. 2026. https://we-calculate.com/calculator/rref-calculator.
TG we-Calculate Editorial Team, "RREF Calculator (Reduced Row Echelon Form)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rref-calculator
@misc{wecalculate_rref_calculator, title = {RREF Calculator (Reduced Row Echelon Form)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rref-calculator}}, year = {2026}, note = {TG we-Calculate} }
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