Pivot¶

Definition¶

A pivot of a nonzero row is the first nonzero entry in that row.

Interpretation¶

A matrix arranges the coefficients into rows and columns; finding the first nonzero entry of a row identifies its pivot position and the corresponding column.

Pivots mark the leading variables and structure of an echelon form.

A 3 by 4 matrix in echelon form with its pivots, the first nonzero entry of each row, circled in columns 1, 3 and 4; column 2 has no pivot and gives a free variable.

Blue columns hold a pivot: $x_1,x_3,x_4$ are basic (pivot) variables; the orange column has none: $x_2$ is a free variable, which can take any value, and the basic variables are then determined by it. The number of pivots is the rank; in echelon form each pivot sits to the right of the one above.

Key Results¶

In row echelon form, each pivot lies to the right of the pivot in the row above.
In reduced row echelon form, every pivot is a leading $1$ and is the only nonzero entry in its column.

Examples¶

In $(0,0,5,2)$ , the pivot is $5$ .
In $(1,3,0)$ , the pivot is $1$ .

Prerequisites¶

Matrix

Sources¶

Math II: Linear Algebra · Week 3

Aliases: pivot element, pivot entry Keywords: leading entry, leading coefficient

Last updated: 20 October 2026 03:02