For given integers $m,n\geq 2$ there are examples of ideals $I$ of complete determinantal local ring $(R, \mathfrak{m}),$ $ \dim R=m+n-1, $ $\mathrm{grade\,}I=n-1$, with the canonical module $\omega_R$ and the property that the socle dimensions of $H_I^{m+n-2}(\omega_R)$ and $H^m_{\mathfrak{m}}(H^{n-1}_I(\omega_R))$ are not finite. In the case of $m=n$, i.e., a Gorenstein ring, the socle dimensions provide further information about the $\tau$-numbers as studied in Mahmood and Schenzel (J. Algebra 372, 56–67, 10). Moreover, the endomorphism ring of $H^{n-1}_I(\omega_R)$ is studied and shown to be an $R$-algebra of finite type but not finitely generated as $R$-module generalizing an example of Schenzel (J. Algebra 344, 229–245, 15).