A matrix $A$ in $(max,min)$-algebra (fuzzy matrix) is called weakly robust if $A^k\otimes x$ is an eigenvector of $A$ only if $x$ is an eigenvector of $A$. The weak robustness of fuzzy matrices are studied and its properties are proved. A characterization of the weak robustness of fuzzy matrices is presented and an $O\left(n^2\right)$ algorithm for checking the weak robustness is described.