Reduced Row Echelon Form Calculator

The reduced row echelon form (RREF) is the unique simplified form of a matrix obtained by Gaussian-Jordan elimination. Every matrix has exactly one RREF — apply the algorithm correctly and you always end up at the same place.

Three properties define RREF:

• Each non-zero row has a leading 1 (called a pivot)
• Each pivot is to the right of pivots in rows above it
• All entries above and below each pivot are zero
• All-zero rows (if any) are at the bottom

This calculator uses Gauss-Jordan with partial pivoting (swap in the largest pivot in absolute value to reduce roundoff error). The algorithm is the cleanest way to solve linear systems, find matrix rank, identify the null space, and compute matrix inverses.

To use RREF for solving Ax = b: form the augmented matrix [A | b], compute RREF. If the last column of [A|b] reduces to a column with only zeros (no pivot in last column), the system is consistent. The pivot columns of A correspond to “basic variables” — solve directly. Free variables (non-pivot columns) parametrize the solution space.

Worked example: A = [[1,2,3],[4,5,6],[7,8,9]] reduces to [[1,0,-1],[0,1,2],[0,0,0]]. Two pivots in columns 1 and 2; column 3 is free. Rank = 2. Nullity = 1. The third row reducing to all zeros tells you the original three rows were linearly dependent — the third row equals 2*(row 2) - (row 1).

Pivot positions (the columns where leading 1s appear) are the columns of A that span its column space. Non-pivot columns are linear combinations of the pivot columns, which the negative entries above each free column reveal directly.

For very small matrices like 2x2 or 3x3, computing RREF by hand is straightforward and a good way to learn the mechanics. The calculator exists for tedious or larger problems and to check your manual work.