The local cohomology modules of thickening of schemes have been studied for decades, and the recent development of representation theory techniques in commutative algebra provides us with powerful tools to understand the structures of the local cohomology and Ext modules of the determinantal thickening. In the case of determinantal thickening of maximal minors and sub-maximal Pfaffians, we employ the aforementioned techniques along with the classification of simple D-modules on spaces of matrices by Raicu, Weyman and Witt to study the module structures of the corresponding Ext modules, thereby answering a question about socle degrees of local cohomology modules raised by Zhang. This is joint work with Mike Perlman.