Augmented Matrix RREF Calculator
Reduce an augmented matrix to Reduced Row Echelon Form with exact fractions, step-by-step row operations, and a shareable result.
RREF Matrix Engine
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How an Augmented Matrix Works
An augmented matrix combines the coefficient matrix of a linear system with the constants on the right side of the equations. It is usually written as [A | b], where A contains the coefficients and b contains the right-hand-side values.
This augmented matrix reduced row echelon form calculator applies Gauss-Jordan elimination to the full matrix, so the final RREF shows whether the system has one solution, no solution, or infinitely many solutions.
How to Enter an Augmented Matrix
Put every equation in the same variable order. Enter the coefficients first, then place the constant from the right side in the final column. A system with n equations and n unknowns becomes an n by n+1 augmented matrix.
2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3
Enter as a 3 by 4 augmented matrix
How to Read the RREF Result
After reduction, the left side identifies pivot variables and free variables. For a unique solution, each variable column has a pivot and the right-hand-side column gives the values.
RREF result
This result reads as x = 2, y = 3, and z = -1. If the final RREF contains a row like [0 0 0 | 5], the system is inconsistent. If a variable column has no pivot and there is no inconsistent row, the system has infinitely many solutions.
Augmented Matrix RREF FAQ
What is an augmented matrix RREF calculator?+
It is a matrix row-reduction calculator for augmented matrices. You enter the coefficient matrix and the right-hand-side column together, then reduce the full matrix to Reduced Row Echelon Form.
How do I enter an augmented matrix?+
Place the coefficients in the first columns and the constants from the right side of the equations in the final column. For example, a system with 3 equations and 3 unknowns is entered as a 3 by 4 matrix.
How do I read the result after RREF?+
A pivot in each variable column gives a unique solution. A row such as 0 = c with c not equal to 0 means no solution. A consistent system with non-pivot variable columns has infinitely many solutions.
Is reduced echelon form the same as reduced row echelon form?+
In linear algebra calculator searches, reduced echelon form usually refers to Reduced Row Echelon Form, or RREF. This page uses RREF as the precise term.