Row Echelon Form Calculator

Row echelon form (REF) is the result of forward Gaussian elimination — applying row operations to make every entry below each pivot zero, working left to right and top to bottom. Unlike reduced row echelon form (RREF), the pivots are not normalized to 1 and entries above pivots are not zeroed out.

REF properties:

• All zero rows (if any) appear at the bottom
• The leading non-zero entry of each non-zero row is to the right of the leading entry in the row above
• Below each leading entry, the column is all zero

REF is not unique — different sequences of row operations can produce different REFs (for example, scaling rows differently). The pivot positions and the rank are uniquely determined, but the actual entries are not.

REF is the natural stopping point if you only need to:

• Determine the rank (count non-zero rows)
• Solve a triangular system by back-substitution
• Compute the determinant (for square matrices, det = product of pivots × (-1)^(row swaps))

Going further to RREF is useful when you want a parametric description of the solution set, the null space basis, or a quick read of which columns are linearly independent. RREF takes more arithmetic but gives more information.

Worked example: A = [[1,2,3],[4,5,6],[7,8,9]]. After elimination: row 2 minus 4row 1 gives [0,-3,-6]. Row 3 minus 7row 1 gives [0,-6,-12]. Row 3 minus 2*(new row 2) gives [0,0,0]. The REF is [[1,2,3],[0,-3,-6],[0,0,0]]. Two pivots, rank 2, det = 0 (third row all zeros means singular).

This calculator uses partial pivoting — at each step it swaps in the row with the largest pivot in absolute value. This reduces roundoff error compared to taking the first non-zero pivot, especially when entries differ by orders of magnitude.

For an upper-triangular result without normalizing pivots, REF is also called the upper-triangular form or the result of “forward elimination.” It is the half-way point between the original matrix and the unique RREF.