  Let $\Lambda$ be an Auslander's 1-Gorenstein Artinian algebra with global
dimension two. If $\Lambda$ admits a trivial maximal 1-orthogonal subcategory
of $\mod\Lambda$, then for any indecomposable module $M \in \mod \Lambda$, we
have that the projective dimension of $M$ is equal to one if and only if so is
its injective dimension and that $M$ is injective if the projective dimension
of $M$ is equal to two. In this case, we further get that $\Lambda$ is a tilted
algebra.
