Let
$n\geq 1$
be an integer and
$f$
be an arithmetical function. Let
$S=\{x_{1},\ldots ,x_{n}\}$
be a set of
$n$
distinct positive integers with the property that
$d\in S$
if
$x\in S$
and
$d|x$
. Then
$\min (S)=1$
. Let
$(f(S))=(f(\gcd (x_{i},x_{j})))$
and
$(f[S])=(f(\text{lcm}(x_{i},x_{j})))$
denote the
$n\times n$
matrices whose
$(i,j)$
-entries are
$f$
evaluated at the greatest common divisor of
$x_{i}$
and
$x_{j}$
and the least common multiple of
$x_{i}$
and
$x_{j}$
, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that
$\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$
, where
$f\ast \unicode[STIX]{x1D707}$
is the Dirichlet convolution of
$f$
and the Möbius function
$\unicode[STIX]{x1D707}$
. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant
$\det (f[S])$
if
$f$
is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants
$\det (f(S\setminus \{1\}))$
and
$\det (f[S\setminus \{1\}])$
. In this paper, we evaluate the determinant
$\det (f(S\setminus \{x_{t}\}))$
for any integer
$t$
with
$1\leq t\leq n$
and also the determinant
$\det (f[S\setminus \{x_{t}\}])$
if
$f$
is multiplicative.