GCD matrix
In mathematics, a  greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix. Definition 

Let  be a list of positive integers. Then the  matrix  having the greatest common divisor  as its  entry is referred to as the GCD matrix on .The LCM matrix  is defined analogously. The study of GCD type matrices originates from  who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the  matrix  is , where  is Euler's totient function. Bourque–Ligh conjecture 
 conjectured that the LCM matrix on a GCD-closed set  is nonsingular. This conjecture was shown to be false by  and subsequently by . A lattice-theoretic approach is provided by . References

Matrix theory
Number theory