I am just posting my comments as one further answer. Every locally complete intersection scheme is Gorenstein (and Cohen-Macaulay). Is a generic determinantal variety Gorenstein? For morphisms from a rank $e$ locally free sheaf to a rank $f$ locally free sheaf and the generic determinantal variety where the rank of the morphism is $\leq r$, Goto proves that the generic determinantal variety is not Gorenstein if both $e\neq f$ and $r\neq 0$, i.e., if it is Gorenstein then either $e$ equals $f$ or $r$ equals $0$ (or both). Also, for $r=1$ and $e=f\leq 5$, Goto proves that the generic determinantal variety is Gorenstein. Here is the reference.

Goto, Shiro
When do the determinantal ideals define Gorenstein rings?.
Sci. Rep. Tokyo Kyoiku Daigaku Sect. A12(1974), 129–145.
https://www.jstor.org/stable/43698822