Determinant of matrix with $A_{ij} = \min (i, j)$

Question

Given a $n\times n$ matrix whose $(i, j)$-th entry is the lower of $i,j$, eg. $$\begin{pmatrix}1 & 1 & 1 & 1\\ 1 & 2 & 2 & 2 \\ 1 & 2 & 3 & 3\\ 1 & 2 & 3 & 4 \end{pmatrix}.$$ The determinant of any such matrix is $1$. How do I prove this? Tried induction but the assumption would only help me to compute the term for $A_{nn}^*$ mirror.

Answer 1

You can substract the $j$-th column to the $(j+1)$-th one. This will leave you with a lower-triangular matrix of all ones.