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Given a matrix $A_{n \times n}$, which has elements $a_{i,j} \sim \mathrm{unif} \left[a,b\right]$, what is the probablity of $\det(A)$ being zero? What if $a_{i,j}$ have any other distribution?

Added: Let's assume an extension of the about problem; What is the probability of, $\mathbb{P}(|\det(A)| < \epsilon ), \; s.t. \; \epsilon \in \mathbb{R} $ ?

Thanks!

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    You need to specify the joint distribution of the $a_{i,j}$ or else taking them all equal gives an example where the probability is one. It is worth noting that asymptotics of the case where $a_{i,j}$ are $iid$ Bernoulli random variables is an open research question with work done by some big names like Bourgain and Vu.2012-10-31
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    This is a nice proof (for the Lebesgue measure, but would seem to generalize to measures that are absolutely continuous wrt the Lebesgue measure) http://www.uwindsor.ca/math/sites/uwindsor.ca.math/files/05-03.pdf However, it is inductive. It would be nice to see other proofs for a result that seems 'obvious'.2012-10-31
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    @Chris, can you be more specific in your comment "taking them all equal" and "probability is one". If the $a_{i,j}$ are $iid$ say uniform in $[0,1]$, then almost surely the determinant is nonzero. Did you mean if the $a_{i,j}$ all have the same value (ie, degenerate distribution)?2012-11-01
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    @alancalvitti: Yes, the point was to show that you need to specify the joint distribution because there's at least one case in which the probability is 1.2012-11-01

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