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Let $R=\mathbb{C}[\{x_{ij}\}]$ be the ring of polynomial functions in $mn$ variables where $m> n$. Set $X$ to be the $m\times n$ matrix in these variables and $I:=I_n(X)$ the ideal of maximal minors of $X$. We consider the rings $R/I^t$; for $t\gg 0$ the depth of $R/I^t$ is equal to $n^2-1$, and we show that each local cohomology module $H^{n^2-1}_{\frak{m}}(R/I^t)$ is a cyclic $R$-module. We also compute the annihilator of $H^{n^2-1}_{\frak{m}}(R/I^t)$ thereby completely determining its $R$-module structure. In the case that $X$ is a $n\times (n-1)$ matrix we describe a map between the Koszul complex of the $t$-powers of the maximal minors and a free resolution of $R/I^t$. We use this map to explicitly describe the modules $\operatorname{Ext}_R ^n(R/I^t,R)$ as submodules of the top local cohomology module $H_I^n(R)$. Moreover, we can realize the filtration $\bigcup_i\operatorname{Ext}_R ^n(R/I^t,R)= H_I^n(R)$ in terms of differential operators. Utilizing this description, along with an explicit isomorphism $H_I^n(R) \cong H_{\frak{m}}^{n(n-1)}(R)$, we determine the annihilator of $\operatorname{Ext}_R ^n(R/I^t,R)$ and hence by graded local duality give another computation of the annihilator of $H^{(n-1)^2-1}_{\frak{m}}(R/I^t)$.