  A sequence of approximations for the determinant and its logarithm of a
complex matrixis derived, along with relative error bounds. The determinant
approximations are derived from expansions of det(X)=exp(trace(log(X))), and
they apply to non-Hermitian matrices. Examples illustrate that these
determinant approximations are efficient for lattice simulations of finite
temperature nuclear matter, and that they use significantly less space than
Gaussian elimination. The first approximation in the sequence is a block
diagonal approximation; it represents an extension of Fischer's and Hadamard's
inequalities to non-Hermitian matrices. In the special case of Hermitian
positive-definite matrices, block diagonal approximations can be competitive
with sparse inverse approximations. At last, a different representation of
sparse inverse approximations is given and it is shown that their accuracy
increases as more matrix elements are included.
