Elementary Row Operation¶

Definition¶

An elementary row operation is one of three reversible operations on a system or matrix: adding a multiple of one row to another, interchanging two rows, or multiplying a row by a nonzero scalar.

Interpretation¶

Elementary row operations are the steps used to solve equations by hand, written for the rows of a matrix: swap two equations, multiply an equation by a nonzero number, or add a multiple of one equation to another. They transform a system without changing its solution set.

Three 3 by 3 matrices before and after an elementary row operation: swapping two rows, scaling a row by a nonzero number, and adding a multiple of one row to another.

Each row of the figure applies one operation to the same matrix. Swapping $\boldsymbol{R}_1$ and $\boldsymbol{R}_2$ only reorders the rows; $\boldsymbol{R}_2\to\tfrac{1}{2}\boldsymbol{R}_2$ turns $(4,5,6)$ into $(2,2.5,3)$ ; and $\boldsymbol{R}_3\to\boldsymbol{R}_3-7\boldsymbol{R}_1$ turns $(7,8,9)$ into $(0,-6,-12)$ , creating a zero below the first entry.

Replace $\boldsymbol{R}_2$ by $\boldsymbol{R}_2+k\boldsymbol{R}_1$ $k R_{1}$ : the solution set stays the same. With $a_{11}\neq0$ $0$ , exact elimination uses $k=-a_{21}/a_{11}$ $- a_{21} / a_{11}$ . The button selects the nearest available control value; compare the applied $k$ and the remaining $x_1$ coefficient to see whether elimination is exact or approximate.

The solutions are kept because every operation can be undone by one of the same kind: a swap by the same swap, scaling by $c$ by scaling by $1/c$ , and adding $k\boldsymbol{R}_1$ to $\boldsymbol{R}_2$ by adding $-k\boldsymbol{R}_1$ , so the two systems are equivalent. The scalar must be nonzero: multiplying a row by $0$ replaces an equation by $0=0$ $0$ , which cannot be undone and can add solutions. Each operation can also be written as multiplication on the left by an elementary matrix, obtained by applying the same operation to the identity matrix.

The same operations drive the simplex pivot in OR Modelling. A common slip is to change two rows at once from the old rows: replacing $\boldsymbol{R}_1$ by $\boldsymbol{R}_1-\boldsymbol{R}_2$ and at the same time $\boldsymbol{R}_2$ by $\boldsymbol{R}_2-\boldsymbol{R}_1$ gives two rows that are negatives of each other, so one equation is lost.

Key Results¶

Elementary row operations preserve equivalence of systems.
Gauss-Jordan elimination consists of applying elementary row operations until reduced row echelon form is reached.

Examples¶

Replace $\boldsymbol{R}_2$ by $\boldsymbol{R}_2-3\boldsymbol{R}_1$ $3 R_{1}$ .
Swap $\boldsymbol{R}_1$ and $\boldsymbol{R}_3$ .

Prerequisites¶

Matrix

Sources¶

Math II: Linear Algebra · Week 3

Aliases: row operation, elementary row transformation

Last updated: 27 October 2026 11:52