Example of putting a matrix in echelon form and identifying the pivot columns

Example: Row-reduce and identify pivot columns:

$$A = \begin{pmatrix} 1 & 3 & 2 & 1 \\ 2 & 6 & 5 & 4 \\ 1 & 3 & 1 & 2 \end{pmatrix}$$

Row reduction:

1. $R_2 \leftarrow R_2 - 2R_1$, $R_3 \leftarrow R_3 - R_1$:
   $$\begin{pmatrix} 1 & 3 & 2 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & -1 & 1 \end{pmatrix}$$
2. $R_3 \leftarrow R_3 + R_2$:
   $$\begin{pmatrix} 1 & 3 & 2 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 3 \end{pmatrix}$$

Pivot columns: Columns 1, 3, and 4 (positions of leading entries in the echelon form). Column 2 is not a pivot column — it corresponds to a free variable.