Solve systems of linear equations by performing real row reduction to reduced row echelon form. Enter a valid augmented matrix and the calculator will show the RREF, rank, and solution.
Last updated: June 2026 | By Patchworkr Team
The left side contains coefficients for x1 through x3. The last column is the constant term.
Gauss-Jordan elimination is a row reduction method that transforms an augmented matrix all the way to reduced row echelon form, or RREF. Once the matrix reaches RREF, the solution can be read directly from the last column when the system has a unique solution.
The method uses three row operations:
Unlike LU-based solving, this calculator shows the actual row reduction process so the matrix transformation matches the math being taught.
What is the difference between Gaussian and Gauss-Jordan elimination?
Gaussian elimination stops at row echelon form. Gauss-Jordan continues to reduced row echelon form so the solution is easier to read.
Why does this calculator reject blanks?
Blank cells are not valid coefficients. Rejecting them avoids silently turning missing input into zero.
What if no unique solution exists?
The calculator will tell you whether the system is inconsistent or has free variables instead of returning a misleading answer.
How many equations can I enter?
This version handles 2x2 through 5x5 systems using an augmented matrix.
Does the calculator show the row operations?
Yes. The row reduction steps are listed under the matrix so you can follow the elimination process.
Can I solve this by hand?
Yes. Perform the same row operations shown here: swap, scale, and eliminate until the matrix reaches RREF.
Why use Gauss-Jordan at all?
It is a direct way to solve systems and also makes the structure of the solution visible, which is useful for learning.
What is RREF?
Reduced row echelon form is a matrix shape where each pivot is 1, every pivot column has zeros elsewhere, and pivots move to the right as you go down the rows.
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