http://oeis.org/A046747
@Mk Utkarsh
yeah i checked that but i didn't find any method or may be i overlooked it.
let X(n) be number of singular matrices of Number of singular $N \times N$ rational (0,1) matrices.
let Y(n) be number of binary matrices of order $N \times N$ whose determinant is exactly zero.
$Y(n) =$$\large 2^{(n^2)} - n! * \binom{2^n -1}{n} + n! * X(n)$
I don't think it is required to to remember all these things for GATE.
actually i thought it also have some trivial method as odd/even #determinant matrices has but someone on SO mentioned it as somehow tough job so yeah may be it wont be asked.
The tests are there but it ain't free. Cost is...