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Generalized GCD matrices. (English) Zbl 1246.11077
The usual Smith type matrix is of the form $[f((x_i, x_j))]$, where the $ij$ entry is the arithmetical function $f$ evaluated at the greatest common divisor $(x_i, x_j)$ of $x_i$ and $x_j$. The present author considers certain analogous matrices of the form $[f(i, (i, j))]$. As an open problem the author asks to evaluate the determinant of this matrix. Solutions to this problem in terms of general meet matrices on semilattices can be found in [{\it B. Lindström}, Proc. Am. Math. Soc. 20, 207--208 (1969; Zbl 0165.02902)] and in [{\it M. Mattila} and {\it P. Haukkanen}, Some properties of row-adjusted meet and join matrices, Linear Multilinear Algebra, in print]. For a general account on this type matrices, see [{\it J. Sándor} and {\it B. Crstici}, Handbook of number theory. II. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1079.11001)].
##### MSC:
 11C20 Matrices, determinants (number theory) 11A25 Arithmetic functions, etc. 15B36 Matrices of integers
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