Find the determinant of an $n \times n$ square matrix $A$ whose entries are $a_{ij} = \max(i,j)$

Question

I figured the matrix would look like this,

$$ A = \begin{bmatrix} 1 & 2 & 3 & \dots & n \\ 2 & 2 & 3 & \dots & n \\ 3 & 3 & 3 & \dots & n \\ \vdots & \vdots & \vdots & \ddots &\vdots \\ n-1 & n-1 & n-1 & \dots & n \\ n & n & n & \dots & n \end{bmatrix} $$

but I do not know how to tackle it. Reduced row echelon row doesn't seem to work here.

Answer 1

As in this answer: if you substract the $(i+1)$-th row to the $i$-th one, you will end up with a lower triangular matrix with all diagonal entries being $-1$ except $A_{nn} = n$, so $\det A = n(-1)^{n-1}$.