On a certain arithmetical determinant
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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 \({x \in S \cup\{1\}}\)). 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