An augmented matrix represents a linear system, so changing its rows must preserve the system's solution set. Three elementary row operations do exactly that:

• swap two rows;
• multiply a row by a nonzero scalar;
• add a multiple of one row to another row.

Each operation corresponds to replacing the equations by equivalent equations.

Eliminating entries

For example,

$$\left[\begin{array}{cc|c} 1&1&5\ 2&-1&1 \end{array}\right]$$

can use

$$R_2\leftarrow R_2-2R_1$$

to obtain

$$\left[\begin{array}{cc|c} 1&1&5\ 0&-3&-9 \end{array}\right].$$

The new system has exactly the same solutions as the old one but is easier to solve.

Row-echelon form

A matrix is in row-echelon form when zero rows are at the bottom, each leading nonzero entry lies to the right of the leading entry above it, and all entries below each leading entry are zero.

The leading entries identify pivot positions. Columns without pivots correspond to free variables when the matrix represents a system.

Reduced row-echelon form

In reduced row-echelon form, each pivot is $1$ and is the only nonzero entry in its column.

Echelon forms expose the structure of a system: which variables are determined, which are free and whether contradictory equations appear.