# 1  Elementary Row Operations and Equivalent Systems of Linear Equations

Given a system of linear equations

 $\begin{array}{ccccccccc}{a}_{1,1}{x}_{1}& +& {a}_{1,2}{x}_{2}& +& \dots & +& {a}_{1,n}{x}_{n}& =& {b}_{1}\\ {a}_{2,1}{x}_{1}& +& {a}_{2,2}{x}_{2}& +& \dots & +& {a}_{2,n}{x}_{n}& =& {b}_{2}\\ ⋮& & & & & & & & ⋮\\ {a}_{m,1}{x}_{1}& +& {a}_{m,2}{x}_{2}& +& \dots & +& {a}_{m,n}{x}_{n}& =& {b}_{m}\end{array}$ (1.1)

We know that the following three types of operations lead to an equivalent system of linear equations:

1. Interchange two equations.
2. Multiply an equation by a nonzero number.
3. Add a multiple of one equation to another equation.

These operations have their counterpart in operations with matrices.

The (matrix) elementary row operations are:

1. Interchange two rows.
2. Multiply a row by a nonzero real number.
3. Add a multiple of one row to another row.

We have the following proposition:

Proposition 1.1. Given a system of linear equations. Denote by ${A}^{R}$ the augmented matrix of the system. Let $C$ be a matrix that was obtained from the matrix ${A}^{R}\phantom{\rule{mediummathspace}{0.2em}}$by one or more elementary row operations. Then $C$ is the augmented matrix of an equivalent system of linear equations.