  Two complex matrix pairs $(A,B)$ and $(A',B')$ are contragrediently
equivalent 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 $(A,B)$ for
contragredient equivalence; that is, a simple normal form to which all matrix
pairs $(A + \widetilde A, B+\widetilde B)$ close to $(A,B)$ can be reduced by
contragredient equivalence transformations that smoothly depend on the entries
of $\widetilde A$ and $ \widetilde B$. Each perturbation $(\widetilde
A,\widetilde B)$ of $(A,B)$ defines the first order induced perturbation
$A\widetilde{B}+\widetilde{A}B$ of the matrix $AB$, which is the first order
summand in the product $(A +\widetilde{A})(B+\widetilde{B}) = AB +
A\widetilde{B}+\widetilde{A}B+ \widetilde A \widetilde B$. We find all
canonical matrix pairs $(A,B)$, for which the first order induced perturbations
$A\widetilde{B}+\widetilde{A}B$ are nonzero for all nonzero perturbations in
the normal form of Garc\'{\i}a-Planas and Sergeichuk. This problem arises in
the theory of matrix differential equations $\dot x=Cx$, whose product of two
matrices: $C=AB$; using the substitution $x = Sy$, one can reduce $C$ by
similarity transformations $S^{-1}CS$ and $(A,B)$ by contragredient equivalence
transformations $(S^{-1}AR,R^{-1}BS)$.
