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Abstract. We show that with any finite partially ordered set, P , one can associate a matrix whose determinant factors nicely. As corollaries, we obtain a number of results in the literature about GCD matrices and their relatives. Our main theorem is proved combinatorially using nonintersecting paths in a directed graph.

We define meet and join matrices on two subsetsXandYof a lattice(P,≼)with respect to a complex-valued functionfonPby(X,Y)f=(f(xi∧yi))and[X,Y]f=(f(xi∨yi)), respectively. We present expressions for the determinant and the inverse of(X,Y)fand[X,Y]f, and as special cases we obtain several new and known formulas for the determinant and the inverse of the usual meet and join matrices(S)fand[S]f.

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