Row-Echelon Form

Using elementary row operations, any non-zero matrix can be reduced to a simplified row-echelon form.

A row all of whose entries are $0$ is called a zero row; a row with at least one non-zero entry is a non-zero row.

Definition 1.5 (row-echelon form). A non-zero matrix $E$ is in row-echelon form if:

1. Every zero row of $E$ occurs below every non-zero row.
2. If the first non-zero entry of row $i$ occurs in column $j$, then every entry below it, in column $j$, is zero.
3. The first non-zero entry of row $i$ lies strictly to the left of the first non-zero entry of row $i+1$.

Informally: all zero rows sit at the bottom, and each row's first non-zero entry (its pivot) is strictly right of the pivot in the row above it.

Method to reduce a matrix to row-echelon form.

Step 1 (pivoting). If the first row is zero, swap it with a non-zero row below. Otherwise, if $a_{11}\ne0$ go to Step 2; if $a_{11}=0$, swap row 1 with a lower row that has a non-zero first-column entry (or, failing that, hunt column-by-column for the first usable pivot). The first non-zero entry secured in row 1 is called the pivot.

Step 2 (clearing below the pivot). Use row operations of the form $R_i\to R_i+\lambda R_1$ to make every entry below the pivot, in the pivot's column, zero.

Step 3 (recurse). Treat the next row as the new "first row" and repeat Steps 1--2 using only the rows below it, until every row has been processed.

Worked illustration. $\begin{pmatrix}0&2&4\\1&1&1\\2&3&1\end{pmatrix}\xrightarrow{R_1\leftrightarrow R_2}\begin{pmatrix}1&1&1\\0&2&4\\2&3&1\end{pmatrix}\xrightarrow{R_3\to R_3-2R_1}\begin{pmatrix}1&1&1\\0&2&4\\0&1&-1\end{pmatrix}\xrightarrow{R_3\to R_3-\frac12R_2}\begin{pmatrix}1&1&1\\0&2&4\\0&0&-3\end{pmatrix}$, a row-echelon form with all three rows non-zero. …