×

On GCD and LCM matrices. (English) Zbl 0761.15013

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 \(1\leq i\), \(j\leq n\).

MSC:

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

Citations:

JFM 08.0074.03
PDF BibTeX XML Cite
Full Text: DOI

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.