TITLE: Determinants of "almost identity" matrices.
QUESTION [5 upvotes]: Suppose that $A$ is a real square matrix with all diagonal entries $1$, all off-diagonal entries non-positive, and all column sums positive and non-zero. Does it follow that $\det(A)\neq0$? Is this just an exercise? Are these matrices well-known?

REPLY [3 votes]: I'm answering to the "are these matrices well-known" part. Yes, they belong to at least two classes of widely studied matrices:

Diagonally dominant matrices, as has been suggested before, i.e., matrices such that $|A_{ii}|>\sum_{j\neq i}|A_{ij}|$ for each $i$.
M-matrices. There are several equivalent definitions of M-matrices, such as matrices in the form $sI-P$, where $P$ is an elementwise nonnegative matrix and $s>\rho(P)$ ($\rho$=spectral radius), or matrices with nonpositive off-diagonal elements and all their eigenvalues in the right half-plane. You can find a comprehensive exposition, including an impressive list of 50 conditions equivalent to "$A$ is a nonsingular M-matrix", on Berman, Plemmons Nonnegative matrices in the mathematical sciences. They have several interesting properties that "look like" those of symmetric positive definite matrices.