Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough.


**What does the distribution of the determinants look like? Is it normal or skewed?**


**What proportion of such $n\times n$ determinants is singular?**



Distribution of $0$-$1$ matrices
Much is known about these Hadamard matrices, this [talk](http://mypage.iu.edu/~worrick/slides/SpectrumANU.pdf) by William Orrick gives a good overview:


The first few moments of the determinant $\det M$ are known exactly, for large $n$ the distribution of $|\det M|$ is conjectured to be log-normal.


For the number of singular matrices, see [OEIS.](http://oeis.org/A046747) The large-$n$ limit of the probability that a $n\times n$ matrix is singular is conjectured to be $n^2/2^n$.


