Let's you have the matrix $A=\begin{pmatrix} \mathbf{1} & 2 & 3 & 4 \\ 5 & \mathbf{6} & 7 & 8 \\ 9 & 10 & 11 & \mathbf{12} \\ 13 & 14 & \mathbf{15} & 16 \\ \end{pmatrix}$.
If you select four entries $a_{ij}$ located on different rows and columns, the sum will always be $34$.
For example, $1 + 6 + 12 + 15 = 34$.
Question. Is it possble to find the sum for the $n \times n$ random matrix with ordered elements before selection $n$ elements?
Edit 1. The element $a_{11}$ can be great that $1$. Edit 2. The elements $a_{ij}$ can be an arithmetic progression.