Why is the rank of a $n \times n$ square matrix of the first positive integers from the smallest until $n^2$ always equal $2$ for $n≥2$?

Question

I tried it out on a $3\times 3$ and a $4\times 4$ matrix and saw that from the third line on all lines are multiples of the second line, so only the first two lines are unique.

E.g. for a $3\times 3$ matrix until $3^2$: $$\left( \begin{matrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{matrix} \right) \Rightarrow \left( \begin{matrix}1 & 2 & 3\\ 0 & 3 & 6\\ 0 & 6 & 12 \end{matrix} \right).$$

However, I don't understand why it is the case and how it would be possible to prove the above statement.

Thank you very much for your help.