Example of putting a matrix in echelon form

Example: Row-reduce the matrix to echelon form:

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{pmatrix}$$

Step 1: $R_2 \leftarrow R_2 - 2R_1$ and $R_3 \leftarrow R_3 - 3R_1$:
$$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$$

Step 2: $R_3 \leftarrow R_3 - R_2$:
$$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$$

This is in row echelon form: nonzero rows are above zero rows, and each leading entry is to the right of the one above it.