Gauss-Jordan procedure to put a matrix into reduced row echelon form

Gauss-Jordan elimination transforms a matrix into reduced row echelon form (RREF).

Algorithm:

1. Forward elimination: Find the leftmost nonzero column. Use row swaps and row operations to create zeros below the pivot. Repeat for each subsequent row.
2. Scale pivots to 1: Divide each row by its pivot value.
3. Backward elimination: Starting from the rightmost pivot, create zeros above each pivot.

Example: For $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \end{pmatrix}$, after backward elimination ($R_1 \leftarrow R_1 - 2R_2$): $\begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \end{pmatrix}$.

The result is unique: every matrix has exactly one RREF.