Abstract
Let S = { x 1, x 2,…, x n } be a set of distinct positive integers. The matrix ( S) having the greatest common divisor ( x i , x j ) of x i and x j as its i, j entry is called the greatest common divisor (GCD) matrix on S. The matrix [ S] having the least common multiple of x i and x j as its i, j entry is called the least common multiple (LCM) matrix on S. The set S is factor-closed if it contains every divisor of each of its elements. If S is factor-closed, we calculate the inverses of the GCD and LCM matrices on S and show that [S](S) −1 is an integral matrix. We also extend a result of H. J. S. Smith by calculating the determinant of [ S] when ( x i , x j )∈ S for 1 ⩽ i, j ⩽ n.
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