Find rank of $A$ with entries $a_{ij}=\min\{i,j\} , 1\leq i,j \leq n $.

Question

Let $A=(a_{ij})$ be a $n\times n$ matrix such that $a_{ij}=\min\{i,j\} , 1\leq i,j \leq n $. Find rank of $A$.

My work :

I think the rank is $n$.

Reason: Let $A_n=(a_{ij}) $. After subtracting the first row from the other rows I get $\det A_n=\det A_{n-1}=\cdots=\det A_1=1$. So $\det A_n=\det A=1$ must be invertible. Hence rank $=n $.

Is my solution correct ? If not provide an alternative solution. Thank you.