## 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

JFM 08.0074.03
Full Text:

### References:

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