MATRIX ROW REDUCTION TOOL
RREF Calculator
Convert a matrix into reduced row echelon form (RREF) using Gauss-Jordan elimination.
View row echelon form, pivot columns, matrix rank, free columns and row-operation steps.
Enter integers, decimals or fractions such as 3/4.
Enter a matrix
Enter one row per line. Separate entries with spaces or commas. Use fractions such as
1/2 if desired. Matrices can have different numbers of rows and columns.
How to use the RREF calculator
- Enter the matrix, using one line per row.
- Separate entries with spaces or commas; fractions such as
3/4are accepted. - Click Calculate RREF.
- Read the REF, RREF, rank, pivot columns and row-operation steps.
What is reduced row echelon form?
A matrix is in RREF when all zero rows are at the bottom, each nonzero row begins with a leading 1, leading 1s move to the right as you move down, and every pivot column has zeros everywhere except at its pivot. RREF is unique for a given matrix.
REF versus RREF
Row echelon form (REF) requires zeros below each pivot. Reduced row echelon form (RREF) goes further by making every pivot equal to one and clearing the entries above each pivot. This calculator shows both forms.
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Frequently asked questions
What does RREF mean?
RREF means reduced row echelon form. It is the unique matrix obtained by applying valid elementary row operations until each pivot is the only nonzero value in its column.
How is matrix rank calculated?
The rank is the number of pivot positions in the row-reduced matrix. It is also the number of nonzero rows in echelon form.
Can I enter a rectangular matrix?
Yes. The calculator accepts rectangular matrices as well as square matrices, with up to 12 rows and 12 columns.
Can I enter fractions?
Yes. Enter fractions like 3/4 or -2/5. The calculator converts them to decimal floating-point values for row reduction, so repeating fractions may be rounded.