  Let $A$ be an $n \times n$ random matrix with iid entries over a finite field
of order $q$. Suppose that the entries do not take values in any additive coset
of the field with probability greater than $1 - \alpha$ for some fixed $0 <
\alpha < 1$. We show that the singularity probability converges to the uniform
limit with an exponentially small error depending only on $\alpha$. We also
show that the distribution of the determinant of $A$ converges to its limiting
distribution at an exponential rate.
