  Using the established $d$-concavity of the $k$-Hessian type functions
$F_k(R)=\log(S_k(R)),$ whose variables are nonsymmetric matrices, we prove $
C^{2, \alpha}(\overline{\Omega}) $ estimates for strictly $(\delta,
\widetilde{\gamma}_k) $-admissible solutions to the Dirichlet problem without
the well-known regularity condition. A necessary condition for the existence of
strictly $\delta$-admissible solutions to the equations is given. By the method
of continuity, we provide some sufficient conditions for the unique solvability
in the class of strictly $(\delta,\widetilde{\gamma}_k)$-admissible solutions
to the Dirichlet problem, provided that those skew-symmetric matrices in the
equations are sufficiently small in some sense.
