Reduced Row Echelon Form Calculator

Transform a 2 by 3 matrix into RREF with Gauss-Jordan elimination and read the solution to a two-variable linear system instantly.

RREF matrix solver
Enter the six values of a 2 by 3 matrix from left to right.

About reduced row echelon form

Reduced row echelon form, usually shortened to RREF, is a standardized version of a matrix that makes its essential information easy to read. A matrix is in RREF when every nonzero row begins with a leading 1, each leading 1 is the only nonzero entry in its column, leading entries move to the right as you move down the rows, and any all-zero rows appear at the bottom. These conditions give every matrix one unique reduced row echelon form, even though different sequences of row operations may be used to reach it. This calculator applies Gauss-Jordan elimination to a two-row, three-column matrix. That shape is especially useful for an augmented system of two linear equations in two unknowns: the first two columns contain coefficients and the final column contains constants. The algorithm searches for a nonzero pivot, swaps rows if necessary, divides the pivot row so its pivot becomes one, and then eliminates every other entry in the pivot column. It repeats the process from left to right until no additional pivot can be formed. Elementary row operations do not change the solution set represented by an augmented matrix. You may swap two rows, multiply a row by any nonzero constant, or add a multiple of one row to another. For example, the matrix with rows [1, 2, 5] and [3, 4, 11] reduces to [1, 0, 1] and [0, 1, 2]. Interpreted as equations, the result states directly that x equals 1 and y equals 2. RREF also reveals systems that do not have one unique solution. A zero row can indicate dependent equations and infinitely many solutions when at least one variable is free. A row whose coefficient entries are zero but whose augmented value is nonzero represents a contradiction and therefore no solution. Beyond equation solving, reduced row echelon form helps determine matrix rank, identify pivot and free columns, test linear independence, and find bases for vector spaces. The displayed values are rounded only for readability; elimination uses full JavaScript floating-point precision.

RREF examples

These matrices illustrate unique, dependent, and already reduced systems.

Input matrixRREFMeaning
[1, 2, 5]; [3, 4, 11][1, 0, 1]; [0, 1, 2]The unique solution is x = 1 and y = 2.
[1, 2, 3]; [2, 4, 6][1, 2, 3]; [0, 0, 0]The second equation is dependent, leaving one free variable.
[1, 0, 4]; [0, 1, -2][1, 0, 4]; [0, 1, -2]The matrix is already reduced and gives x = 4 and y = -2.

How to calculate RREF

  1. Enter each matrix value in its labeled row and column position.
  2. Use the third column for constants when representing two linear equations.
  3. Select Calculate RREF to perform Gauss-Jordan elimination.
  4. Read pivots, zero rows, and solution values from the reduced matrix.

RREF calculator FAQ

What is the difference between REF and RREF?

Row echelon form requires zeros below each pivot, while reduced row echelon form also requires zeros above every pivot. RREF therefore exposes solutions more directly and is unique for a given matrix.

Can RREF solve simultaneous equations?

Yes, enter the coefficients and constants as an augmented matrix. A pivot in each variable column gives a unique solution that can be read from the final column.

What does a row of zeros mean?

A zero row means one original equation did not add independent information. If a variable column has no pivot, the consistent system has at least one free variable and infinitely many solutions.

How does the calculator handle decimal values?

It performs elimination with double-precision floating-point numbers. Results are cleaned near zero and displayed with up to eight decimal places for readability.

What row operations are allowed?

You may swap rows, multiply a row by a nonzero number, or add a multiple of one row to another. Each operation preserves the solution set of the represented linear system.