This paper studies irreducible matrices $A=(a_{ij})\in\mathbb C^{n\times n}$, $n\ge2$, satisfying Brualdi's conditions $$ \prod_{i\in\overline\gamma}|a_{ii}|\ge\prod_{i\in\overline\gamma}R_i(A), \quad \gamma\in\mathfrak C(A), $$ or, shortly, Brualdi matrices. Here, $R_i(A)=\sum\limits_{i\ne j}|a_{ij}|$, $i=1,\dots,n$; $\mathfrak C(A)$, is the set of circuits of length $k\ge2$ in the directed graph of $A$, and $\overline\gamma$ is the support of $\gamma$. Among the results obtained are a characterization of Brualdi's matrices, implying, in particular, that they are generalized diagonally domiant; necessary and sufficient conditions of singularity for Brualdi matrices; explicit expressions for the absolute values of the components of right null-vectors of a singular Brualdi matrix, and conditions necessary and sufficient for a boundary point of Brualdi's inclusion region to be an eigenvalue of an irreducible matrix.