What is Reduced Row Echelon Form (RREF)?

Reduced Row Echelon Form (RREF) is a standardized form of a matrix resulting from Gauss-Jordan elimination where each leading entry (pivot) is 1, each pivot is to the right of the pivot above, all entries in pivot columns are zero except the pivot itself, and all-zero rows are at the bottom.

Can this RREF calculator handle exact fractions?

Yes! Our calculator computes all row operations using exact rational fractions (like 1/3, -5/7) instead of floating-point approximations, guaranteeing mathematical accuracy.

How does the augmented matrix solver work for systems of equations?

In augmented mode [A|b], the last column represents constant terms. The calculator checks rank to identify if the linear system has a unique solution, infinitely many solutions with free variables, or no solution (inconsistent).

TI 84 CalculatorFree Simulator

TI-84 TI-30XS Casio RREF

Linear Algebra & Matrix Suite•Gauss-Jordan Elimination

Reduced Row Echelon Form Calculator

Find the reduced row echelon form (RREF) or row echelon form (REF) of any matrix with step-by-step row operations, exact fraction arithmetic, and linear system solutions.

Rows:

Cols:

Presets:

Reduced (RREF)(Jordan)Augmented [A|b]

Enter Matrix Coefficients & Constants:Accepts fractions (e.g. 1/3, -4/5) and decimals

Presets:

Calculated Result

Reduced Row Echelon Form (RREF)

1	0	0	2
0	1	0	-1
0	0	1	3

System Solution:Unique Solution

The system has a unique solution.

x_1 =2

x_2 =-1

x_3 =3

Matrix Rank

Nullity / Free Vars

Pivot Columns

C1, C2, C3

Row Operations

Step-by-Step Gauss-Jordan Elimination

Detailed elementary row operations with exact fractional arithmetic

Step 1Subtract 2 times row 1 from row 2

R_{2} ← R_{2} - 2 × R_{1}

1	2	3	9
0	-5	-5	-10
3	0	-1	3

Step 2Subtract 3 times row 1 from row 3

R_{3} ← R_{3} - 3 × R_{1}

1	2	3	9
0	-5	-5	-10
0	-6	-10	-24

Step 3Multiply row 2 by -1/5 to make the leading entry 1

R_{2} ← -\frac{1}{5} × R_{2}

1	2	3	9
0	1	1	2
0	-6	-10	-24

Step 4Subtract 2 times row 2 from row 1

R_{1} ← R_{1} - 2 × R_{2}

1	0	1	5
0	1	1	2
0	-6	-10	-24

Step 5Add 6 times row 2 to row 3

R_{3} ← R_{3} + 6 × R_{2}

1	0	1	5
0	1	1	2
0	0	-4	-12

Step 6Multiply row 3 by -1/4 to make the leading entry 1

R_{3} ← -\frac{1}{4} × R_{3}

1	0	1	5
0	1	1	2
0	0	1	3

Step 7Subtract 1 times row 3 from row 1

R_{1} ← R_{1} - R_{3}

1	0	0	2
0	1	1	2
0	0	1	3

Step 8Subtract 1 times row 3 from row 2

R_{2} ← R_{2} - R_{3}

1	0	0	2
0	1	0	-1
0	0	1	3

Why Use Our RREF Matrix Calculator?

Designed specifically for linear algebra students, engineers, and researchers requiring transparent, step-by-step matrix solutions.

Exact Fractional Steps

Never get confused by imprecise decimals. Every single elementary row operation uses exact rational fractions (like 2/3, -5/7), so your coursework remains clean and accurate.

Linear System Solver

Enable the augmented column [A|b] to automatically classify the underlying system of linear equations: unique solution, infinite solutions with free variables, or inconsistent.

Rank & Nullity Analysis

Instantaneous computation of matrix rank, nullity, pivot columns, and row operation counts. Copy LaTeX brackets directly into your homework or Overleaf documents.

How Gauss-Jordan Elimination Works

Four systematic phases to transform any matrix into Reduced Row Echelon Form.

Find Pivot

Locate the leftmost non-zero entry in the current working row. Swap rows if necessary.

Scale to 1

Divide or multiply the pivot row by a constant scalar so that the leading entry becomes exactly 1.

Clear Column

Add or subtract multiples of the pivot row to turn all other entries in the pivot column into 0.

Repeat Down

Move to the next submatrix row and repeat until every pivot is isolated and zero rows sit at the bottom.

Frequently Asked Questions

Common questions about matrix rank, pivots, and linear algebra solutions.