What Is a Gauss-Jordan Elimination Calculator?
A Gauss-Jordan elimination calculator reduces a matrix to reduced row echelon form (RREF) using elementary row operations, which solves systems of linear equations, finds matrix inverses, and determines rank. Gauss-Jordan elimination extends Gaussian elimination all the way to a fully reduced form. Enter your matrix and the calculator returns the RREF with steps.
How to Use the Calculator
- Enter your matrix — the coefficients (and constants for a system).
- Calculate — see the reduced row echelon form and the solution.
What Reduced Row Echelon Form Is
In RREF, each leading entry (pivot) is 1, each pivot is the only nonzero entry in its column, and the pivots step down to the right. From RREF, the solution of a linear system can be read directly.
The Three Row Operations
| Operation | Description |
|---|---|
| Swap | Exchange two rows |
| Scale | Multiply a row by a nonzero constant |
| Add | Add a multiple of one row to another |
These operations do not change the solution set, so the RREF represents the same system in its simplest form.
Gauss-Jordan vs Gaussian Elimination
Gaussian elimination reduces a matrix to row echelon form (upper triangular) and then uses back substitution. Gauss-Jordan goes further, continuing until the matrix is in fully reduced form, so the solution can be read off directly without back substitution.
What You Can Solve
- Linear systems: read solutions directly from RREF.
- Matrix inverse: augment with the identity and reduce.
- Rank: count the nonzero rows in RREF.
- Consistency: spot no-solution or infinite-solution cases.
Frequently Asked Questions
What is Gauss-Jordan elimination?
It is a method that uses row operations to reduce a matrix to reduced row echelon form, directly solving linear systems and finding inverses.
What is reduced row echelon form?
RREF is a matrix form where each pivot is 1 and is the only nonzero entry in its column, with pivots stepping to the right, allowing solutions to be read off directly.
How is it different from Gaussian elimination?
Gaussian elimination stops at row echelon form and uses back substitution, while Gauss-Jordan continues to fully reduced form so no back substitution is needed.
What are the allowed row operations?
Swapping two rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another — none of which change the solution.
Is this Gauss-Jordan calculator free?
Yes — it is completely free, requires no signup, and returns the RREF with steps.