Determinant of matrices with entries $a_{i, j} = \operatorname{gcd}(i, j)$

Question

Suppose we have $n \times n$ matrix $A$ with $a_{i,j}={\rm gcd}(i,j)$. What is the determinant of $A$?

Answer 1

This quantity is the so-called Smith determinant and turns out to be $$\det A = \prod_{k = 1}^n \varphi(k),$$ where $\varphi$ is Euler's totient function. Smith's original paper is

H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.

but, despite its age, it is gated. A modern (and ungated) explanation, which exploits the LU decomposition, appears, e.g., in

Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009), pp. 43-49.

More references appear in A001088, the OEIS entry for the sequence $$1, 1, 2, 4, 16, 32, 192, 768, \ldots,$$ whose $n$th entry is the determinant $\det A$ of the $n \times n$ matrix.

Answer 2

The result is $$ \det(A)=\prod_{k=1}^n\phi(k). $$ see here.