Gauss–Jordan Elimination Calculator
The Gauss Jordan calculator accepts integer, decimal or fraction entries, computes exact RREF, lists elementary row operations, identifies pivots and classifies an optional augmented linear system.
Gauss–Jordan Elimination Calculator inputs
The Gauss Jordan calculator accepts integer, decimal or fraction entries, computes exact RREF, lists elementary row operations, identifies pivots and classifies an optional augmented linear system.
Gauss–Jordan Elimination Calculator results
How to use the gauss jordan calculator
The Gauss Jordan calculator accepts integer, decimal or fraction entries, computes exact RREF, lists elementary row operations, identifies pivots and classifies an optional augmented linear system. This focused guide explains the exact inputs, conversions and limitations so the output can be audited instead of treated as a generic black box.
Exact reduced row echelon form
A Gauss Jordan calculator transforms a matrix until every nonzero row has a leading 1, each leading 1 is the only nonzero entry in its column, pivot positions move right as rows move down and all-zero rows appear at the bottom. Those conditions define reduced row echelon form. This engine uses rational arithmetic for every operation, so entries such as one third remain exact fractions rather than becoming rounded decimals that can change a pivot decision.
Accepted matrix entries
Enter one matrix row per line and separate entries with spaces or commas. Integers, terminating decimals and fractions are accepted. The Gauss Jordan calculator converts decimals to exact base-ten fractions before elimination. Every row must contain the same number of entries, and the browser solver limits the matrix size to keep the full operation history readable. Invalid denominators, ragged rows and empty matrices are rejected with a specific message.
Recorded elementary row operations
The result lists row swaps, row scaling and row replacement operations in execution order. That history makes the Gauss Jordan calculator useful for checking hand work rather than merely displaying a final matrix. A row may be swapped into the pivot position, divided by its leading value and then used to eliminate the same column from every other row. Equivalent operation sequences can produce the same unique RREF, so a textbook may show different intermediate steps.
Rank and pivot columns
Rank equals the number of pivot columns in the reduced matrix. Nonpivot columns correspond to free variables when the matrix is a coefficient matrix. The Gauss Jordan calculator reports pivot positions using one-based column numbers and verifies the number of nonzero RREF rows. Rank is a structural property of the matrix; it is not the same as the number of original rows or columns unless the matrix has full row or column rank.
Augmented-system classification
When the last column is marked as augmented, the calculator separates coefficient pivots from the right-hand side. A row with all zero coefficients and a nonzero augmented entry represents an inconsistent equation, so the system has no solution. If every variable column is a pivot and the system is consistent, the solution is unique. Otherwise the system has infinitely many solutions. The displayed RREF supports writing the parameterized solution explicitly.
Limits and verification
Large sparse systems, symbolic parameters and ill-conditioned measured data may be better handled in specialist linear-algebra software. Exact fraction numerators can also grow quickly. This Gauss Jordan calculator is intended for classroom matrices and transparent checks. Verify that the entered matrix matches the problem, retain the operation list for auditing and avoid interpreting a rounded measurement as mathematically exact merely because it was typed as a decimal.
Gauss–Jordan Elimination Calculator questions
Is RREF unique?
Yes. Every matrix has one unique reduced row echelon form.
Why are decimals shown as fractions?
They are converted to exact base-ten fractions to avoid floating-point pivot errors.
What makes an augmented system inconsistent?
A reduced row with zero coefficients and a nonzero right-hand side.