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In this note we describe the singular locus of diagonally-dominant Hermitianmatrices with nonnegative diagonal entries over the reals, the complex numbers,and the quaternions. This yields explicit expressions for the probability thatsuch matrices, chosen at random, are singular. For instance, in the case of theidentity $n \times n$ matrix perturbed by a symmetric, zero-diagonal, $\{\pm1/(n-1)\}$-Bernoulli matrix, this probability turns out to be equal to$2^{-(n-1)(n-2)/2}$. As a corollary, we find that the probability for a$n\times n$ symmetric $\{\pm 1\}$-Bernoulli matrix to have eigenvalue $n$ is$2^{-(n^2-n+2)/2}$.

The singularity probability of random diagonally-dominant Hermitian matrices