On a certain arithmetical determinant

Abstract

Smith showed in 1875 that if ${n \geqq 1}$ is an integer and ${G := {({\rm gcd}(i, j))}_{1 \leqq i, j \leqq n}}$ is the ${n \times n}$ matrix having gcd(i, j) as its i, j-entry for all integers i and j between 1 and n, then ${{\rm det}(G) = {\prod_{k=1}^{n}}\varphi(k)}$, where ${\varphi}$ is the Euler’s totient function. We show that if ${n \geqq 2}$ is an integer and ${H := {({\rm gcd}(i, j))}_{2 \leqq i, j \leqq n}}$ is the ${(n-1) \times (n-1)}$ matrix having gcd(i, j) as its i, j-entry for all integers i and j between 2 and n, then

$${\rm det}(H) = (\prod \limits_{k=1}^{n} \varphi(k)) \sum \limits_{\begin{array}{c} k=1 \\ k {\rm is squarefree} \\ \end{array}}^{n} \frac{1}{\varphi(k)}.$$

We also calculate the determinants of the matrices ${{(f({\rm gcd}(x_{i}, x_{j})))}_{1 \leqq i, j \leqq n}}$ and ${{(f({\rm lcm}(x_{i},x_{j})))}_{1\leqq i, j\leqq n}}$ having f evaluated at ${{\rm gcd}(x_{i}, x_{j})}$ and ${{\rm lcm}(x_{i}, x_{j})}$ as their (i, j)-entries, respectively, where ${S = \{x_{1},\ldots, x_{n}\}}$ is a set of distinct positive integers such that ${x_{i} > 1}$ for all integers i with ${1 \leqq i \leqq n}$ and ${S \cup \{1\}}$ is factor closed (that is, ${S \cup \{1\}}$ contains every divisor of x for any ${x \in S \cup\{1\}}$). Our result answers partially an open problem raised by Ligh [18].