  Let $f$ be an arithmetical function. The matrix $[f(i,j)]_{n\times n}$ given
by the value of $f$ in greatest common divisor of $(i,j)$, $f\big((i,j)\big)$
as its $i,\; j$ entry is called the greatest common divisor (GCD) matrix. We
consider the generalization of this matrix where the elements are in the form
$f\big(i,(i,j)\big)$.

Number Theory