Elementary Row Operations
This page follows the JoeCMath video Master Matrix Elementary Row Operations in 5 Minutes. The video introduces the three row moves used when working with matrices, especially when preparing for Gaussian elimination or Gauss-Jordan elimination.
Main Idea
In the video, elementary row operations are treated as three controlled ways to rewrite whole rows of a matrix:
- Swap two rows.
- Multiply one row by a nonzero scalar.
- Add a multiple of one row to another row.
In row notation, those moves are written as:
\[R_i \leftrightarrow R_j\] \[cR_i \to R_i, \quad c\ne 0\]and
\[R_i+cR_j \to R_i.\]The important habit is to watch which row is being replaced. In the notation $R_i+cR_j\to R_i$, row $i$ changes, but row $j$ is only being used to help create the new row.
Watch this section: elementary row operations at 0:00.
Swapping Two Rows
The first operation in the video is a row swap. Swapping rows means two rows trade positions, and no arithmetic is needed.
For example, the video uses a matrix like:
\[\begin{bmatrix} 3 & 6 & 9 \\ 0 & 3 & 6 \\ -2 & 4 & 8 \end{bmatrix}\]Swapping row $1$ and row $3$ is written as:
\[R_1 \leftrightarrow R_3.\]So:
\[\begin{bmatrix} 3 & 6 & 9 \\ 0 & 3 & 6 \\ -2 & 4 & 8 \end{bmatrix} \xrightarrow{R_1 \leftrightarrow R_3} \begin{bmatrix} -2 & 4 & 8 \\ 0 & 3 & 6 \\ 3 & 6 & 9 \end{bmatrix}\]Only rows $1$ and $3$ move. Row $2$ stays in the middle exactly as it was.
Watch this section: swap two rows at 0:20.
Multiplying a Row by a Nonzero Scalar
The second operation changes one row by multiplying every entry in that row by the same nonzero number.
In the video, the row operation:
\[\frac{1}{3}R_1 \to R_1\]turns the first row into:
\[\frac{1}{3} \begin{bmatrix} 3 & 6 & 9 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}.\]So the matrix changes like this:
\[\begin{bmatrix} 3 & 6 & 9 \\ -2 & 4 & 8 \\ 0 & 3 & 6 \end{bmatrix} \xrightarrow{\frac{1}{3}R_1 \to R_1} \begin{bmatrix} 1 & 2 & 3 \\ -2 & 4 & 8 \\ 0 & 3 & 6 \end{bmatrix}\]The scalar must be nonzero. Multiplying by $0$ would erase a row, which is not one of the allowed elementary row operations.
Watch this section: multiply a row by a nonzero scalar at 1:08.
Adding a Multiple of One Row to Another
The third operation is the one that does most of the work in row reduction. You take a multiple of one row, add it to another row, and replace only the target row.
The video uses the operation:
\[R_1+2R_2 \to R_1.\]Start with:
\[\begin{bmatrix} 1 & 2 & 3 \\ -2 & 4 & 8 \\ 0 & 3 & 6 \end{bmatrix}\]Now calculate the new first row:
\[R_1+2R_2 = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} +2 \begin{bmatrix} -2 & 4 & 8 \end{bmatrix} = \begin{bmatrix} -3 & 10 & 19 \end{bmatrix}.\]That gives:
\[\begin{bmatrix} 1 & 2 & 3 \\ -2 & 4 & 8 \\ 0 & 3 & 6 \end{bmatrix} \xrightarrow{R_1+2R_2 \to R_1} \begin{bmatrix} -3 & 10 & 19 \\ -2 & 4 & 8 \\ 0 & 3 & 6 \end{bmatrix}\]Rows $2$ and $3$ stay the same. The arrow tells you that row $1$ is the row being replaced.
Watch this section: add a multiple of one row to another row at 2:37.
All Three Operations Together
The review at the end of the video collects the three operations into one short list.
| Operation | Notation | What it means |
|---|---|---|
| Swap two rows | $R_i \leftrightarrow R_j$ | Rows $i$ and $j$ trade places. |
| Scale one row | $cR_i \to R_i$, where $c\ne 0$ | Every entry in row $i$ is multiplied by the same nonzero number. |
| Add rows | $R_i+cR_j \to R_i$ | A multiple of row $j$ is added to row $i$, and row $i$ is replaced. |
These are the row moves you use when turning a matrix into row echelon form or reduced row echelon form. The notation is compact, but the idea is always row-by-row: identify the target row, do the arithmetic, and leave the other rows alone unless the operation says otherwise.
Watch this section: review all three row operations at 4:17.
Timestamp Guide
Related Linear Algebra Work
After these row operations feel comfortable, the next natural companion page is Row Echelon Form, where the video checks the final shape of a matrix using zero rows and pivot positions.
For more JoeCMath matrix videos, use the Matrix/Linear Algebra playlist.