Gauss-Jordan Elimination Calculator

Gauss-Jordan elimination is an algorithm for solving systems of linear equations. It extends Gaussian elimination by continuing row operations until the matrix is in reduced row echelon form (RREF), where each pivot is 1 and all other entries in the pivot column are 0 — directly yielding the solution without back-substitution.

A matrix is in RREF when: (1) all zero rows are at the bottom, (2) the leading entry (pivot) of each nonzero row is 1, (3) each pivot is to the right of the pivot in the row above, and (4) each pivot is the only nonzero entry in its column. RREF uniquely represents the solution of the system.

Infinitely many solutions occur when the rank of the coefficient matrix is less than the number of unknowns and the system is consistent. In this case, some variables are 'free' — they can take any value — and the remaining variables are expressed in terms of those free variables.

A system has no solution (is inconsistent) when row reduction produces a row of the form [0 0 … 0 | c] where c ≠ 0. This represents the impossible equation 0 = c, indicating the equations contradict each other.

An augmented matrix combines the coefficient matrix of a linear system with the constant column on the right-hand side, separated by a vertical bar. For example, the system 2x + y = 5, x − y = 1 becomes the augmented matrix [2 1 | 5; 1 -1 | 1].

The rank is the number of pivot positions in the RREF of the matrix. It equals the maximum number of linearly independent rows (or columns). A system of n unknowns has a unique solution if and only if the rank of the augmented matrix equals n and equals the rank of the coefficient matrix.

Enter the augmented matrix row by row, with values separated by spaces. Separate rows using semicolons. For example, a 3×3 system would look like: '2 1 -1 8; -3 -1 2 -11; -2 1 2 -3'. Each row must contain (number of unknowns + 1) values — coefficients for each variable followed by the right-hand side constant.

Gaussian elimination reduces a matrix to row echelon form (upper triangular), after which back-substitution is needed to find the solution. Gauss-Jordan elimination goes further, reducing to RREF by also eliminating entries above each pivot, so the solution is read off directly without any back-substitution step.