  In this article we show that the symbolic Rees algebra of a mixed ladder
determinantal ideal is strongly $F$-regular. Furthermore, we prove that the
symbolic associated graded algebra of a mixed ladder determinantal ideal is
$F$-pure. The latter implies that mixed ladder determinantal rings are
$F$-pure. We also show that ideals of the poset of minors of a generic matrix
give rise to $F$-pure algebras with straightening law.
