  We establish a criterion for the strong $F$-regularity of a (non-Gorenstein)
Cohen-Macaulay reduced complete local ring of dimension at least $2$,
containing a perfect field of prime characteristic $p$. We also describe an
explicit generating morphism (in the sense of Lyubeznik) for the top local
cohomology module with support in certain ideals arising from an $n\times
(n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to
derive a simple, new proof of the well-known fact that the generic
determinantal ring defined by the maximal minors of $X$ is strongly
$F$-regular.
