⬛ RREF Calculator

RREF Calculator

Reduced row echelon form (RREF) is the unique simplified form of a matrix reached by Gaussian elimination, and this calculator shows every row operation along the way. Supports augmented matrices for solving linear systems.

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Row Reduction Calculator

Gauss-Jordan Elimination with full steps

✓ Step-by-Step✓ Augmented✓ Free⚡ Loading precision engine…
Matrix A
Coefficient Matrix A
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Constants b
Quick Examples

What is RREF?

The Reduced Row Echelon Form (RREF) of a matrix is a unique form obtained by performing elementary row operations. A matrix is in RREF when:

  • The leading entry (pivot) in each non-zero row is 1
  • Each pivot is to the right of pivots in rows above
  • All entries above and below each pivot are 0
  • Zero rows are at the bottom

Why Use RREF?

RREF is used to solve systems of linear equations, find the rank of a matrix, determine linear independence, and find null spaces and column spaces.

Elementary Row Operations

  • Swap: Exchange two rows → multiplies det by −1
  • Scale: Multiply a row by a non-zero constant
  • Replace: Add a multiple of one row to another

Worked Example

For the matrix [[2,4],[1,3]]: after row reduction, it becomes [[1,0],[0,1]] — the identity matrix. This tells us the matrix is full rank and, if it came from a system of equations, that system has exactly one solution.

What Does RREF Tell You?

Once a matrix is in RREF, you can immediately read off the rank (the number of non-zero rows), see whether a linear system has one solution, many, or none, and identify which variables are "free" versus determined.

Frequently Asked Questions

Row Echelon Form (REF) only requires zeros below each pivot. Reduced Row Echelon Form (RREF) goes further, requiring zeros both above and below each pivot, and each pivot must equal 1. RREF is unique for a given matrix; REF is not.
Yes — unlike regular row echelon form, there is only one possible RREF for any given matrix, no matter which order you perform the row operations in.
Each row represents one equation. A row like [1, 0, | 3] means that variable equals 3 directly. A row of all zeros means that equation added no new information.

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