Determinant value of a square matrix whose each entry is the g.c.d. of row and column position

Question

Let $A=(a_{ij})$ be a $n \times n$ matrix with $a_{ij}=\gcd(i,j) , \forall i,j=1,2, \cdots, n$ , then how do we prove $\det A=\prod_{i=1}^n \phi(i)$ ? , where $\phi$ is the Euler's phi function

Answer 1

This is called gcd matrix and Smith determinant, you can find proofs here.

One of the ideas is that, through basic matrix operation, you can transform the given matrix into a triangular one with $\phi(i)$ in the diagonal line.