It is known that the stability of a Metzler matrix can be characterized in a Routh-Hurwitz-like fashion based on a recursive application of scalar Schur complements [1]. Our objective in this brief note is to show that recently obtained stability conditions are equivalent statements of this result and can be deduced directly there from using only elementary results from linear algebra. Implications of this equivalence are also discussed and several examples are given to illustrate potentially interesting system-theoretic applications of this observation.