The usual Smith type matrix is of the form , where the entry is the arithmetical function evaluated at the greatest common divisor of and . The present author considers certain analogous matrices of the form . 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 [B. Lindström, Proc. Am. Math. Soc. 20, 207–208 (1969; Zbl 0165.02902)] and in [M. Mattila and 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 [J. Sándor and B. Crstici, Handbook of number theory. II. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1079.11001)].