Matrices associated with multiplicative functions

Abstract Let f be a multiplicative function and S = { x 1 , x 2 , …, x n } a set of distinct positive integers. Denote by ( f [ x i , x j ]) the n × n matrix having f evaluated at the least common multiple [ x i , x j ] of x i and x j as its i , j entry. If S is factor-closed, we calculate the determinant of this matrix and (if it is invertible) its inverse, and show that for a certain class of functions the n × n matrix ( f ( x i , x j )) having f evaluated at the greatest common divisor of x i and x j as its, i , j entry is a factor of the matrix ( f [ x i , x j ]) in the ring of n × n matrices over the integers. We also determine conditions on f that will guarantee the matrix ( f [ x i , x j ]) is positive definite.