Row operations

Definition: Row operations (elementary row operations) are three fundamental operations that can be performed on the rows of a matrix without changing the solution set of the associated linear system:

1. Row swap: Swap two rows: $R_i \leftrightarrow R_j$
2. Row scaling: Multiply a row by a nonzero scalar: $R_i \leftarrow cR_i$ where $c \neq 0$
3. Row replacement: Add a multiple of one row to another: $R_i \leftarrow R_i + cR_j$ where $i \neq j$

Example: Starting with:
$$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 1 & 1 \end{pmatrix}$$
Applying $R_2 \leftarrow R_2 - 2R_1$:
$$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}$$

Row operations are used in Gaussian elimination to transform a matrix into row echelon form or reduced row echelon form (RREF).