  The averaging method combined with the Lyapunov-Schmidt reduction provides
sufficient conditions for the existence of periodic solutions of the following
class of perturbative $T$-periodic nonautonomous differential equations
$x'=F_0(t,x)+\varepsilon F(t,x,\varepsilon)$. Such periodic solutions bifurcate
from a manifold $\mathcal{Z}$ of periodic solutions of the unperturbed system
$x'=F_0(t,x)$. Determining the stability of this kind of periodic solutions
involves the computation of eigenvalues of matrix-valued functions
$M(\varepsilon)$, which can done using the theory of $k$-hyperbolic matrices.
Usually, in this theory, a diagonalizing process of $k$-jets of
$M(\varepsilon)$ must be employed and no general algorithm exists for doing
that. In this paper, we develop an alternative strategy for determining the
stability of the periodic solutions without the need of such a diagonalization
process, which can work even when the diagonalization is not possible.
Applications of our result for two families of $4$D vector fields are also
presented.
