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 {\bf a} = (a_1, \cdots, a_n) non-negative real numbers a_n>0 $, give sufficient conditions on continuous function H:\mathbb{R}\times\mathbb{R}_0^+\to\mathbb{R} in order viscosity supersolutions of$ \mathcal{P}_{\bf a}(D^2u) H(u, |Du|) $in co...