Matrix Product Parity

Matrix Parity Probability
ProbabilityMedium

Given a matrix S where each entry can only be either 1 or -1, how many different matrices can we construct such that:

  1. The product of all numbers in each row equals 1
  2. The product of all numbers in each column equals 1

Further: If we randomly generate a matrix with +1 and -1 elements only, what's the probability the we get such a matrix?