![Sample our Mathematics & Statistics journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5943/TOC-Subject-Sample-2021-Mathematics-Statistics.jpg"})
Abstract
Let a, b and h be positive integers and S = {x
1, … , x
h
} be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation σ of {1, … , h} such that x
σ(1)|…|x
σ(h). We say that the set S consists of two coprime divisor chains if we can partition S as S = S
1 ∪ S
2, where S
1 and S
2 are divisor chains and each element of S
1 is coprime to each element of S
2. The matrix having the ath power (x
i
, x
j
)
a
of the greatest common divisor (GCD) of x
i
and x
j
as its (i,j)-entry is called the ath power GCD matrix defined on S, denoted by (S
a
). Similarly, we can define the ath power least common multiple (LCM) matrix [S
a
]. In the first paper of the series, Tan [Q. Tan, Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains, Linear Multilinear Algebra 58 (2010), pp. 659--671] showed that if S consists of two coprime divisor chains and 1 ∈ S and a|b, then (S
a
)|(S
b
), [S
a
]|[S
b
] and (S
a
)|[S
b
] hold in the ring M
h
(Z) of h × h matrices over integers. But such factorizations need not hold if a ∤ b. In this second paper of the series, we assume that S consists of two coprime divisor chains and 1 ∉ S. We show the following results: (i) If , then
,
and
. (ii) If a|b, then det(S
a
) | det(S
b
), det[S
a
] | det[S
b
] and det(S
a
) | det[S
b
]. (iii) If a|b, then (S
a
) | (S
b
), [S
a
] | [S
b
] and (S
a
) | [S
b
] hold in the ring M
h
(Z) if and only if both
and
are integers, where S = S
1 ∪ S
2 with S
1 and S
2 divisor chains and x = min(S
1) and y = min(S
2). Our results extend Hong’s results and complement Tan’s results.
Acknowledgements
The authors thank the anonymous referee for helpful comments. This research was supported partially by the Research Fund of Panzhihua University.