pr.probability – Distribution and Expectation of Inverse of a Random Bernoulli Matrix

This question cropped up as a part of my research. Let us assume a $ntimes n$ random matrix $mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values ${0,1}$ with probability $ p = 1/2$.

What I want to know is that what sort of distribution would $mathbf{M}^{-1}$ have? and what could its possible expectation be?

My guess so far has been that the distribution remains the same. I have posted this question on math.stackexchange too.