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Two complex matrix pairs $(A,B)$ and $(A',B')$ are contragredientlyequivalent if there are nonsingular $S$ and $R$ such that$(A',B')=(S^{-1}AR,R^{-1}BS)$. M.I. Garc\'{\i}a-Planas and V.V. Sergeichuk(1999) constructed a miniversal deformation of a canonical pair $(A,B)$ forcontragredient equivalence; that is, a simple normal form to which all matrixpairs $(A + \widetilde A, B+\widetilde B)$ close to $(A,B)$ can be reduced bycontragredient equivalence transformations that smoothly depend on the entriesof $\widetilde A$ and $ \widetilde B$. Each perturbation $(\widetildeA,\widetilde B)$ of $(A,B)$ defines the first order induced perturbation$A\widetilde{B}+\widetilde{A}B$ of the matrix $AB$, which is the first ordersummand in the product $(A +\widetilde{A})(B+\widetilde{B}) = AB +A\widetilde{B}+\widetilde{A}B+ \widetilde A \widetilde B$. We find allcanonical matrix pairs $(A,B)$, for which the first order induced perturbations$A\widetilde{B}+\widetilde{A}B$ are nonzero for all nonzero perturbations inthe normal form of Garc\'{\i}a-Planas and Sergeichuk. This problem arises inthe theory of matrix differential equations $\dot x=Cx$, whose product of twomatrices: $C=AB$; using the substitution $x = Sy$, one can reduce $C$ bysimilarity transformations $S^{-1}CS$ and $(A,B)$ by contragredient equivalencetransformations $(S^{-1}AR,R^{-1}BS)$.

Perturbation analysis of a matrix differential equation $\dot x=ABx$