  The purpose of this paper is to compute the asymptotics of determinants of
finite sections of operators that are trace class perturbations of Toeplitz
operators. For example, we consider the asymptotics in the case where the
matrices are of the form $ (a_{i-j} \pm a_{i+j+1-k})_{i,j=0... N-1} $ with $k$
is fixed. We will show that this example as well as some general classes of
operators have expansions that are similar to those that appear in the Strong
Szeg\"{o} Limit Theorem. We also obtain exact identitities for some of the
determinants that are analogous to the one derived independently by Geronimo
and Case and by Borodin and Okounkov for finite Toeplitz matrices. These
problems were motivated by considering certain statistical quantities that
appear in random matrix theory.
