I have a question on determining the minimum number of dependent columns (of the maximum number of independent ones) of a matrix when the coefficients are all modulo $m\in \mathbb{N}$. For example, take this matrix where the coefficients are modulo $13$: $$H=\begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 3 & 5 & 7 & 9 & 11 \\ 1 & 9 & 12 & 10 & 3 & 4 \\ 1 & 1 & 8 & 5 & 1 & 5 \end{pmatrix}$$
I want to know what is the least number of dependent columns I can find. Since there are $4$ rows, I know that the maximum number of independent columns is $4$ (correct me if I'm wrong) and therefore the least number of dependent ones would be $5$, so I would just have to calculate all the possible $4\times 4$ determinants and check if they are non-zero. However, I am wondering if the fact that the top row has all $1$'s gives me some information on the problem. Since the first element is equal for all the columns, can I conclude that every $4$ columns are linearly independent? I am not sure about this, beacuse as long as I take a combination of columns like the following: $$a_1C_1+\ldots a_jC_j\quad \mbox{where} \quad a_1+\ldots+a_j\equiv_{13}1 \quad \mbox{and} \quad j<5$$ then I could get that this sum is another column. Can somenone help me here?