In this paper, we consider two extreme sets of zero-term rank sum of fuzzy matrix pairs: $$cal{z}_1(cal{F})={(X,Y){in}cal{M}_{m,n}(cal{F})^2{mid}z(X+Y)=min{z(X),z(Y)}};$$ $$cal{z}_2(cal{F})={(X,Y){in}cal{M}_{m,n}(cal{F})^2{mid}z(X+Y)=0}$$. We characterize the linear operators that preserve these two extreme sets of zero-term rank sum of fuzzy matrix pairs.