Suppose $A$ is a square matrix of order $n \geq 4$, and $a\_{ij} = i+j$ represents the entry in row $i$ and column $j$. What is the rank of $A$? So we have by the rank-nullity theorem,

$$n = \text{rank}(A) + \text{nullity}(A)$$

For example, for $n=4$, we have $$A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \\ 5 & 6 & 7 & 8 \end{bmatrix}$$ 

So it seems that every row shares elements with the other rows. Can we use this to find the nullity of $A$ and hence the rank of $A$?