GCD matrix

Let ${\displaystyle S=(x_{1},x_{2},\ldots ,x_{n})}$ be a list of positive integers. Then the ${\displaystyle n\times n}$ matrix ${\displaystyle (S)}$ having the greatest common divisor ${\displaystyle \gcd(x_{i},x_{j})}$ as its ${\displaystyle ij}$ entry is referred to as the GCD matrix on ${\displaystyle S}$.The LCM matrix ${\displaystyle [S]}$ is defined analogously.[1][2]

The study of GCD type matrices originates from Smith (1875) who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the ${\displaystyle n\times n}$ matrix ${\displaystyle (\gcd(i,j))}$ is ${\displaystyle \phi (1)\phi (2)\cdots \phi (n)}$, where ${\displaystyle \phi }$ is Euler's totient function.[3]

Bourque & Ligh (1992) conjectured that the LCM matrix on a GCD-closed set ${\displaystyle S}$ is nonsingular.[1] This conjecture was shown to be false by Haukkanen, Wang & Sillanpää (1997) and subsequently by Hong (1999).[4][2] A lattice-theoretic approach is provided by Korkee, Mattila & Haukkanen (2019).[5]