What is the rank of
\(\begin{bmatrix} 1 & -2 & 0 & -1\\1 & 0 & 1 & 1 \end{bmatrix}\)?
The answer is 2.
The rank is given by the dimension of the column space (or the row space).
One can of course row reduce the matrix and then count pivots.
But since each column is a 2-tuple, the dimension can't exceed 2.
Note that columns 2 and 3 are linearly independent.
Hence, the column space must have dimension at least 2.
It follows that the column space has dimension exactly 2
and so the rank is 2.