Pivot vs Pivot Column

A simple introduction to concept of pivots and pivot columns in linear algebra.

Main Idea

In row echelon form, a pivot is the first nonzero entry in a nonzero row from left to right.

Once a pivot is found, the whole column containing that pivot is called a pivot column.

Watch this section: introduction at 0:00.

Finding Pivots in Row Echelon Form

This is a matrix in row echelon form:

\[\begin{bmatrix} 3 & 1 & -1 & 5 \\ 0 & 2 & 2 & 1 \\ 0 & 0 & 6 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix}.\]

The pivots are the first nonzero entries in the nonzero rows:

\[3 \ (\text{in row one, column one}),\quad 2 \ (\text{in row two, column two}),\quad 6 \ (\text{in row three, column three}).\]

Identifying pivots are essential in classify whether a matrix is in row echelon form or reduced row echelon form.

Watch this section: finding pivots in row echelon form at 0:08.

Finding Pivot Columns in Row Echelon Form

Pivot columns are the columns that contain pivots.

For the matrix above, the pivots appear in columns $1$, $2$, and $3$. So the pivot columns are

\[\begin{bmatrix} 3 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ 2 \\ 0 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} -1 \\ 2 \\ 6 \\ 0 \end{bmatrix}.\]

Column $4$ is not a pivot column, even though it has nonzero entries:

\[\begin{bmatrix} 5 \\ 1 \\ 4 \\ 0 \end{bmatrix}.\]

No row has its first nonzero entry from left to right in column $4$.

Watch this section: finding pivot columns in row echelon form at 0:34.

The Same Difference in Reduced Row Echelon Form

In reduced row echelon form, pivots are leading ones.

A pivot column still means the whole column containing one of those pivots.

For example, in a reduced row echelon form matrix like

\[\begin{bmatrix} 1 & 0 & -2 & 4 \\ 0 & 1 & 3 & -5 \\ 0 & 0 & 0 & 0 \end{bmatrix},\]

the pivots are the entries $1$ in positions $(1,1)$ and $(2,2)$.

The pivot columns are column $1$ and column $2$:

\[\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \quad\text{and}\quad \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}.\]

The entries $-2$, $4$, $3$, and $-5$ are not pivots because they are not the first nonzero entries in their rows from left to right.

Watch this section: finding pivots in reduced row echelon form at 0:56 and finding pivot columns in reduced row echelon form at 1:11.

Timestamp Guide

Section	What is shown	Video
Introduction	The video sets up the difference between one pivot entry and a whole pivot column.	0:00
Pivots in row echelon form	Locate the first nonzero entry in each nonzero row.	0:08
Pivot columns in row echelon form	Turn each pivot position into the full column that contains it.	0:34
Pivots in reduced row echelon form	Use the same pivot idea after the matrix is reduced.	0:56
Pivot columns in reduced row echelon form	Identify the full columns containing the reduced pivots.	1:11

Related Linear Algebra Work

This video is easiest to place between the Row Echelon Form companion and the Reduced Row Echelon Form companion.

You can also browse the Matrices topic page or use the JoeCMath Matrix/Linear Algebra playlist.