Summary: In this paper we study a Strong Minimum Principle of Vázquez type for partial trace operators with gradient terms. More explicitly, given an \(n\)-tuple \(\mathbf{a}=(a_1,\cdots,a_n)\) of non-negative real numbers with \(a_n>0\), we give sufficient conditions on a continuous function \(H:\mathbb{R}\times\mathbb{R}_0^+\to\mathbb{R}\) in order for non-negative viscosity supersolutions of \[\mathcal{P}_{\mathbf{a}}(D^2u)=H(u,|Du|)\] in connected open subsets of \(\mathbb{R}^n\) that vanish at some point in \(\Omega\) to vanish identically in \(\Omega\). When \(H\) depends only on the gradient, the condition is also necessary. Here \(\mathcal{P}_{\mathbf{a}}\) 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 \(\mathbf{a}=(a_1,\cdots,a_n)\), both a Strong Maximum Principle and a Compact Support Principle for subsolutions will also be investigated. Not only does our work cover a new class of degenerate operators and a wide class of Hamiltonians not investigated in the literature, but to the best of our knowledge, some of our results are new even when \(\mathcal{P}_{\mathbf{a}}\) reduces to the standard Laplacian.