Row Echelon Form
This page follows the JoeCMath video Row Echelon Form Has Only TWO Rules. The video focuses on recognizing whether a matrix is already in row echelon form, using zero rows and pivot positions.
Main Idea
The video uses two conditions for row echelon form:
- All rows that contain only zeroes are at the bottom of the matrix.
- The leftmost nonzero entry in any row, called a pivot in the video, is to the right of the pivot in the row above it.
The lesson does not define row echelon form by asking for leading ones or reduced columns. It stays focused on the shape of the matrix: zero rows sink to the bottom, and pivots move to the right as you go down.
Watch this section: row echelon form conditions at 0:00.
Rule 1: Zero Rows Go at the Bottom
The first example checks a matrix where one row is all zeroes:
\[\begin{bmatrix} 3 & 1 & -1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 7 & 0 & 2 & 6 \\ 0 & 0 & 1 & 2 & 3 \end{bmatrix}\]The problem is not the zero row itself. A zero row is allowed in row echelon form.
The problem is its location. The zero row is in row $2$, but rows $3$ and $4$ still contain nonzero entries. Since all zero rows need to be at the bottom, this matrix is not in row echelon form.
Watch this section: matrix example without zeroes in the bottom row at 0:17.
Rule 2: Pivots Move to the Right
The next example fixes the zero-row problem but still fails row echelon form:
\[\begin{bmatrix} 3 & 1 & -1 & 5 & 0 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 7 & 0 & 2 & 6 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\]Condition $1$ is satisfied because the all-zero row is at the bottom.
Now look at the leftmost nonzero entry in each nonzero row. Row $2$ has its pivot in column $3$:
\[\begin{bmatrix} 0 & 0 & 1 & 2 & 3 \end{bmatrix}\]Row $3$ has its pivot in column $2$:
\[\begin{bmatrix} 0 & 7 & 0 & 2 & 6 \end{bmatrix}\]That pivot moved left instead of right, so condition $2$ fails. This matrix is not in row echelon form.
Watch this section: matrix example without cascading pivots at 0:39.
A Matrix That Is in Row Echelon Form
The video then shows the same entries arranged so both rules work:
\[\begin{bmatrix} 3 & 1 & -1 & 5 & 0 \\ 0 & 7 & 0 & 2 & 6 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\]The zero row is at the bottom.
The pivots also move to the right:
- row $1$ has pivot $3$ in column $1$,
- row $2$ has pivot $7$ in column $2$,
- row $3$ has pivot $1$ in column $3$.
That creates the stair-step shape the video highlights. Since both conditions are satisfied, the matrix is in row echelon form.
Watch this section: matrix example that is in row echelon form at 1:28.
Practice Examples in the Video
After the main examples, the video shows several matrices and checks the two rules one at a time.
For example, this matrix is in row echelon form:
\[\begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 8 \\ 0 & 0 & 9 \end{bmatrix}\]There are no zero rows above nonzero rows, and the pivots move from column $1$ to column $2$ to column $3$.
This matrix is also in row echelon form:
\[\begin{bmatrix} 5 & -1 & 8 & 4 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\]The zero row is at the bottom, and the pivot in row $2$ is to the right of the pivot in row $1$.
A useful habit from this part of the video is to check condition $1$ first. If a zero row appears above a nonzero row, the matrix fails immediately. If condition $1$ is fine, then scan the pivots from top to bottom.
Watch this section: determining if examples are in row echelon form at 1:51.
Common Ways the Shape Can Fail
The ending of the video shows a few scenarios that prevent row echelon form.
One problem is stacked pivots. If two rows have pivots in the same column, the lower pivot is not to the right of the pivot above it.
For example:
\[\begin{bmatrix} 2 & 1 & 3 \\ 4 & 0 & 7 \\ 0 & 2 & 3 \end{bmatrix}\]The first two rows both start with nonzero entries in column $1$, so the pivots do not move right.
Another problem is placing a zero row above a nonzero row:
\[\begin{bmatrix} 0 & 0 & 0 \\ 4 & 0 & 7 \\ 0 & 2 & 3 \end{bmatrix}\]That fails condition $1$ because a row of all zeroes appears before rows that still contain nonzero entries.
Watch this section: scenarios where a matrix is not in row echelon form at 3:26.
Timestamp Guide
Related Linear Algebra Work
For another matrix-focused Linear Algebra topic, use the cofactor expansion companion guide.
The video description also links to the JoeCMath Gauss-Jordan elimination playlist: watch the playlist on YouTube.