Reduction from SAT to binary matrix subset problem
The original problem asks for a given $k$ whether there is a solution with this $k$, so we can choose to instantiate $k=m$. This means we are forced to pick $m$ columns.
Then if you choose both $x_i$ and $\neg x_i$, there will be a $j$ so that the line number $j$ is left with only zeros, so this choice cannot be good.