<p>This thesis is focused on determinantal rings in 2 different contexts. In Chapter 3 the homological properties of powers of determinantal ideals are studied. In particular the focus is on local cohomology of determinantal thickenings and we explicitly describe the $R$-module structure of some of these local cohomology modules. In Chapter 4 we introduce \textit{generalized diagonal} matrices, a class of sparse matrices which contain diagonal and upper triangular matrices. We study the ideals of minors of such matrices and describe their properties such as height, multiplicity, and Cohen-Macaulayness. </p>