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?



Determinants of "almost identity" matrices.
A nonzero row vector $v$ has a nonzero coordinate of greatest magnitude. $vA$ has a nonzero entry in that coordinate by the triangle inequality, hence is not the $0$ vector.


