The purpose of this dissertation is to study the strong minimum/maximum principle and to investigate various Louisville-type theorems for partial trace equations with nonlinear gradient terms. To investigate the problems and obtain the results presented in this dissertation, we will use various principles (minimum and maximum principles, a comparison principle for viscosity sub/super solutions, boundary principles, and compact support principles), concepts of duality, and different mathematical estimates and inequalities. In this study, we examine a strong minimum principle of Vazquez type for partial trace operators with gradient terms. More explicitly, given a n-tuple a = (a1, · · · , an) of non-negative real numbers with an > 0, we give sufficient conditions on a continuous function h: R × R+0 → R in order for non-negative viscosity super solutions of Pa(D2u) = h(u, |Du|) (⋆) in connected open subsets of Rn that vanish at some point to Ω vanish identically in Ω. When h depends only on the gradient, the condition is also necessary. Here Pa belongs to a class of fully nonlinear degenerate elliptic operators that includes the Min-Max operator which is defined as the sum of the minimum and the maximum eigenvalues of the Hessian matrix. Under suitable conditions on h and a = (a1, · · · , an), both the Strong Maximum Principle and the Compact Support Principle for sub solutions are also investigated. The work covers a new class of degenerate operators and a wide class of Hamiltonian's not investigated in the literature and some of the results are new even when Pa reduces to the standard Laplacian. Illustrative examples are presented for such equations. We investigate various Louisville-type theorems for partial trace equations with nonlinear gradient terms. Specifically, we establish sufficient conditions for which v their viscosity sub solutions vanish identically in Rn. For a prototype of such equations, that is, when h(u, |Du|) = f(u) + g(u)|Du|q for 0 < q < 2 in (⋆), we give necessary and sufficient conditions for viscosity sub solutions identically zero in Rn. Our results are important in the field of mathematical analysis. They serve as a fundamental tool for analyzing the existence and uniqueness, regularity, and blow-up estimate behavior of solutions for practical problems.