On GCD and LCM matrices. (English) Zbl 0761.15013

Bourque, Keith; Ligh, Steve

Linear Algebra Appl. 174, 65-74 (1992).

Authors’ summary: Let \(S=\{x_1,x_2,\dots,x_n)\) 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 [Proc. Lond. Math. Soc. 7, 208–212 (1875; JFM 08.0074.03] by calculating the determinant of \([S]\) when \((x_i,x_j)\in S\) for \(1\leq i\), \(j\leq n\).

Reviewer: M. E. Keating (London)

MSC:

15B36	Matrices of integers
11C20	Matrices, determinants in number theory
15A15	Determinants, permanents, traces, other special matrix functions

Keywords:

greatest common divisor matrix; least common multiple matrix; integral matrix; determinant

Citations:

JFM 08.0074.03

References:

[1]	Apostol, T. M., Arithmetical properties of generalized Ramanujan sums, Pacific J. Math., 41, 281-293 (1972) · Zbl 0226.10045
[3]	Beslin, S.; Ugh, S., Greatest common divisor matrices, Linear Algebra Appl., 118, 69-76 (1989) · Zbl 0672.15005
[4]	Beslin, S.; Ligh, S., Another generalization of Smith’s determinant, Bull. Austral. Math. Soc., 40, 3, 413-415 (1989) · Zbl 0675.10002
[7]	Li, Z., The determinants of GCD matrices, Linear Algebra Appl., 134, 137-143 (1990) · Zbl 0703.15012
[8]	McCarthy, P. J., A generalization of Smith’s determinant, Canad. Math. Bull., 29, 109-113 (1988) · Zbl 0588.10005
[9]	Niven, I.; Zuckerman, H. S., (An Introduction to the Theory of Numbers (1980), Wiley: Wiley New York)
[10]	Smith, H. J.S., On the value of a certain arithmetical determinant, (Proc. London Math. Soc., 7 (1875-1876)), 208-212 · JFM 08.0074.03