Reduced Row Echelon Form Calculator

The reduced row echelon form (RREF) is the unique simplified form of a matrix produced by Gauss-Jordan elimination. Every matrix has exactly one RREF. Apply the algorithm correctly and you always land in the same place, no matter which order you chose the row operations in. That uniqueness is what separates RREF from plain row echelon form, where a different sequence of operations gives different numbers.

Four properties define RREF:

• Each non-zero row has a leading 1, called a pivot
• Each pivot sits to the right of the pivots in the 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: at each step it swaps in the row with the largest entry in that column, which limits floating-point damage from dividing by a small pivot. Because RREF is unique, pivoting changes the arithmetic on the way but never the answer.

Solving Ax = b. Form the augmented matrix [A | b] by writing b as one extra column, and give this calculator all of it. Type 1,2,3,6;4,5,6,15;7,8,10,25 and it recognises the extra column as the right-hand side. If the last column ends up holding a pivot, one row reads 0 = a non-zero number, and the system has no solution. Otherwise the pivot columns of A are the basic variables and the remaining columns are free variables that parametrise 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, and column 3 is free. Rank = 2, nullity = 1. The third row collapsing to zeros says the original three rows were linearly dependent, and here the third equals 2×(row 2) - (row 1).

That same reduced form hands you the null space for free. Column 3 is free, and the entries above it read -1 and 2, so the vector (1, -2, 1) satisfies Ax = 0. Flip the sign of each free-column entry, put a 1 in the free slot, and you have a basis vector. Do that for every free column and you have the whole null space.

For a 2x2 or a 3x3, working RREF by hand is quick and worth doing once for the mechanics. The calculator is for the tedious cases and for checking work you have already done.