Matrices associated with multiplicative functions

AbstractLet f be a multiplicative function and S = {x1, x2, …, xn} a set of distinct positive integers. Denote by (f[xi, xj]) the n × n matrix having f evaluated at the least common multiple [xi, xj] of xi and xj 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(xi, xj)) having f evaluated at the greatest common divisor of xi and xj as its, i, j entry is a factor of the matrix (f[xi, xj]) in the ring of n × n matrices over the integers. We also determine conditions on f that will guarantee the matrix (f[xi, xj]) is positive definite