  We study the low temperature limit of lattice QCD by using a reduction
formula for a fermion determinant. The reduction formula, which is useful in
finite density lattice QCD simulations, contains a reduced matrix defined as
the product of $N_t$ block-matrices. It is shown that eigenvalues of the
reduced matrix follows a scaling law with regard to the temporal lattice size
$N_t$. The $N_t$ scaling law leads to two types of expressions of the fermion
determinant in the low temperature limit; one is for small quark chemical
potentials, and the other is for larger quark chemical potentials.
