Reduced Row Echelon Form

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Transform a matrix into RREF using Gaussian elimination steps.

Input 3x3 Matrix

Resulting RREF

Matrix Simplified

RREF Result

Simplified matrix where leading ones are isolated.

1.00

-1.00

1.00

2.00

ConsistentYes

Rank3

About this calculator

Quick Reference

Leading EntryAlways 1

PivotsStaggered Right

All Zero RowsBottom Level

What is RREF?

Reduced Row Echelon Form (RREF) is a specific type of matrix resulting from Gaussian elimination. To be in RREF, a matrix must be in Row Echelon Form (REF) and also satisfy the condition that every leading 1 is the only non-zero entry in its column. This form is extremely useful for solving systems of linear equations and finding matrix rank.

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Pro Tips

Process: Use elementary row operations: swap rows, multiply rows by constants, or add a multiple of one row to another.
Solution: If the matrix is an augmented matrix for a system of equations, RREF often gives the solution directly (e.g., x=2, y=3).
Rank: The number of non-zero rows in RREF is the rank of the matrix.

Fun Facts

"Gaussian elimination is named after Carl Friedrich Gauss, though Chinese mathematicians used similar methods in 'The Nine Chapters on the Mathematical Art' in 179 AD."
"Calculators and software use LU decomposition or RREF for most internal linear algebra solvers."

Formula: Matrix → RREF