I presume you mean invertible in $M_{n\times n}(\mathbb{Q})$. Here's an example. $$\left(\begin{matrix} 1&2&3&4&5&6\\ 2&1&3&4&5&6\\ 1&3&2&4&5&6\\ 3&1&2&4&5&6\\ 2&3&1&4&5&6\\ 3&2&1&4&5&6 \end{matrix}\right)$$ The last three columns are linearly dependent.

I presume you mean invertible in $M_{n\times n}(\mathbb{Q})$. Here's an example. $$\left(\begin{matrix} 1&2&3&4&5&6\\ 2&1&3&4&5&6\\ 1&3&2&4&5&6\\ 3&1&2&4&5&6\\ 2&3&1&4&5&6\\ 3&2&1&4&5&6 \end{matrix}\right)$$ The last three columns are linearly dependent.

A slightly more interesting question is whether it's true for a transitive (and hence regular) subgroup of order $n$.