Abstract: The authors first obtained formulae for the determinants of
reciprocal power GCD matrix and reciprocal power LCM matrix defined
on a set $S$ consisting of three quasi-coprime divisor chains with
$\gcd (S)\in S$. Consequently, it was showed that the reciprocal
power GCD matrix and the reciprocal power LCM matrix defined on $S$
with $\gcd (S)\in S$ are nonsingular. But such result is not true
when $S$ consists of three divisor chains which are not
quasi-coprime.

Key words: three quasi-coprime divisor chains, greatest-type divisor, reciprocal power GCD matrix, reciprocal power LCM matrix