Row Echelon Form Calculator

What is Row Echelon Form?

Row Echelon Form (REF) is a matrix form obtained through Gaussian elimination where:

All zero rows (if any) are at the bottom
The first non-zero element (leading entry) of each non-zero row is 1 (called a pivot)
Each pivot is to the right of the pivot in the row above it
Entries below pivots are zeros

Row echelon form is fundamental in linear algebra because it:

A matrix is in row echelon form if it satisfies these conditions:

\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 5 & 6 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \end{bmatrix}

In this example:

Number of Rows ($m$)

Number of Columns ($n$)

Set matrix dimensions and click "Generate Matrix" to begin.

Enter matrix values: -

No matrix generated yet. Set dimensions and click "Generate Matrix".

The matrix is now in row echelon form. All pivots have been created and zeros are below each pivot.

You can continue to Reduced Row Echelon Form (RREF) by making zeros above pivots, but that's beyond the scope of REF.