Computing the rank of a matrix

Elimination. $L_2 \leftarrow L_2 - 2L_1$:
$$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix}.$$
$L_3 \leftarrow L_3 - L_2$:
$$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}.$$

Conclusion. There are two nonzero rows, so $\operatorname{rank}(A) = 2$.