Say a $5\times 5$ matrix $$A = \left[ \begin{array}{ccc} 1&2&3&4&5\\ 6&7&8&9&10\\ 11&12&13&14&15\\ 16&17&18&19&20\\ 21&22&23&24&25\\ \end{array} \right] $$
or $n\times n$ in general, is there a quick way of finding the rank of this kind of matrix without using any row operations?
Thank you very much!
With this particularly special matrix, you can observe that if you look at a particular row $r$, then it is the average of row $r - i$ and row $r + i$ (provided $r - i \geq 1$ and $r + i \leq 5$, so that the rows we're talking about are actually there). More generally, though not necessary, is that row $a$ + row $d$ = row $b$ + row $c$ for $1 \leq a \leq b \leq c \leq d \leq 5$ such that $a + d = b + c$.
This tells you that all the intermediate rows are linear combinations of the first and last row, which are visibly linearly independent since neither is a multiple of the other. Hence, $\text{rank}(A) = 2$.