RREF Calculator

Reduced row echelon form is how systems of linear equations get solved systematically: leading 1s, zeros everywhere else in their columns. Paste any matrix — one row per line — and this calculator performs Gauss–Jordan elimination with partial pivoting and reports the RREF and the rank.

Separate entries with commas or spaces, e.g. “1, 2” then “3, 4” on the next line

Results

Formula

Gauss–Jordan elimination uses three legal row operations — swap rows, scale a row, add a multiple of one row to another — until every pivot is 1 and the only nonzero entry in its column:
Row ops never change the solution set of the system

How to use this calculator

  1. Type or paste the matrix with one row per line.
  2. Keep the same number of entries in every row.
  3. Press Calculate to see the RREF and the rank.
RREF Calculator — how the calculation works, illustrated
RREF Calculator — how the calculation works, illustrated

Worked example

Matrix with rows (1, 2) and (3, 4).

  1. Eliminate below the first pivot, then normalize.
  2. Eliminate above the second pivot.
  3. The result is the identity matrix — the system has a unique solution.

Result: RREF is the 2×2 identity matrix; rank = 2.

Frequently asked questions

What is the difference between REF and RREF?

Row echelon form only clears below each pivot. Reduced REF also clears above each pivot and scales pivots to exactly 1, giving a unique final form.

What does rank mean?

The rank is the number of pivots — equivalently the number of independent rows (or columns). For a system, rank tells you whether solutions exist and how many free variables there are.

How do I solve a system with RREF?

Append the constants as an extra column (an augmented matrix). In RREF, each row reads directly as one variable’s value — or reveals free variables when there are more columns than pivots.

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