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Singularity of random Bernoulli matrices. (English) Zbl 1458.15023

Summary: For each \(n\), let \(M_n\) be an \(n\times n\) random matrix with independent \(\pm 1\) entries. We show that \(\mathbb{P}\{M_n\text{ is singular}\}=(1/2+o_n(1))^n\), which settles an old problem. Some generalizations are considered.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15B52 Random matrices (algebraic aspects)
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