Let $A_n$ be the real $n × n$ matrix $(n ≥ 2)$ whose $(i, j)$ entry is $i − j$. What is the rank of $A_n$ as a function of $n$?
My attempt:- $A_2$ has rank $2$. It is obvious. $A_3$ has also rank $2$. Since, $A_3$ has determinant $0$ and has a submatrix of order $2$ with determinant non-zero. Similarly, I could conclude that $A_4$ has rank $2$. Using this method I can't go beyond. I also know that $A_n$ is a skew-symmetric matrix. $\det A_n=0,\forall n.$ I am not able to draw conclusion beyond this. Please help me.