Definition:Elementary Matrix Operation

Definition

Elementary Row Operation

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.

The elementary row operations on $\mathbf A$ are operations which act upon the rows of $\mathbf A$ as follows.

For some $i, j \in \closedint 1 m: i \ne j$:

 $(\text {ERO} 1)$ $:$ $\ds r_i \to \lambda r_i$ For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$ $(\text {ERO} 2)$ $:$ $\ds r_i \to r_i + \lambda r_j$ For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$ $(\text {ERO} 3)$ $:$ $\ds r_i \leftrightarrow r_j$ Exchange rows $i$ and $j$

Elementary Column Operation

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.

The elementary column operations on $\mathbf A$ are operations which act upon the columns of $\mathbf A$ as follows.

For some $i, j \in \closedint 1 n: i \ne j$:

 $(\text {ECO} 1)$ $:$ $\ds \kappa_i \to \lambda \kappa_i$ For some $\lambda \in K_{\ne 0}$, multiply column $i$ by $\lambda$ $(\text {ECO} 2)$ $:$ $\ds \kappa_i \to \kappa_i + \lambda \kappa_j$ For some $\lambda \in K$, add $\lambda$ times column $j$ to column $i$ $(\text {ECO} 3)$ $:$ $\ds \kappa_i \leftrightarrow \kappa_j$ Interchange columns $i$ and $j$

Also defined as

The order of presentation of the elementary matrix operations, either row or column, may vary according to the source.

Some sources use the Greek letter $\rho$ to enumerate the rows, and $\kappa$ to enumerate the columns, and jocularly remind us that the name rho of the letter $\rho$ is pronounced row.

Also see

• Results about elementary matrix operations can be found here.