Most time-optimal parallel algorithms to calculate the determinant and inverse of a matrix
The class NC is the class of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. Is your problem to decide - or actually compute the Determinant of a Matrix? Chistov's algorithm solves for the determinant in the class NC with $(log$ $n)^2$ processors, and is the best(?) so far for the class NC(Reference:  Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic. A. L. Chistov ). As an aside, the question whether the class  P is equivalent to NC is still open.