4.6 Elementary Row Operations on a Matrix

4.6 Elementary Row Operations on a Matrix

Usually, a given system of linear equations is reduced to a simple equivalent system by applying elementary operations which are stated as below:

(i) Interchanging two equations.
(ii) Multiplying an equation by a non-zero number.
(iii) Adding a multiple of one equation to another equation.

Corresponding to these three elementary operations, the following elementary row operations are applied to matrices to obtain equivalent matrices:

(i) Interchanging two rows
(ii) Multiplying a row by a non-zero number
(iii) Adding a multiple of one row to another row

Notations that are used to represent row operations for (i) to (iii) are given below:

• Interchanging $R_i$ and $R_j$ is expressed as $R_i \leftrightarrow R_j$.
• $k$ times $R_i$ is denoted by $kR_i \rightarrow R'_i$.
• Adding $k$ times $R_j$ to $R_i$ is expressed as $R_i + kR_j \rightarrow R'_i$ ($R'_i$ is the new row obtained after applying the row operation).

For equivalent matrices $A$ and $B$, we write $A \cong B$.
If $A \cong B$ then $B \cong A$.

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Note
Matrices $A$ and $B$ are equivalent if $B$ can be obtained by applying in turn a finite number of row operations on $A$.