Geršgorin variations IV: A left eigenvector approach

We use the left vanishing eigenvector to prove various well-known conditions for determining the nonsingularity of matrices via row sums. This is in contrast to the classical approach of using the right vanishing eigenvector. We show that on occasion this approach results in simpler proofs and generalizations of well-known results. We also present a simple proof of a generalized Gudkov's theorem.