  In this paper, we prove that the generic link of a generic determinantal ring
defined by maximal minors is strongly $F$-regular. In the process, we
strengthen a result of Chardin and Ulrich in the graded setting. They showed
that the generic residual intersections of a complete intersection ring with
rational singularities again have rational singularities. We show that if the
said complete intersection is defined by homogeneous elements and is
$F$-rational, then in fact, its generic residual intersections are strongly
$F$-regular in positive prime characteristic.
  Hochster and Huneke showed that determinantal rings are strongly $F$-regular;
however, their proof is quite involved. Our techniques allow us to give a new
and simple proof of the strong $F$-regularity of determinantal rings defined by
maximal minors.
