The paper is devoted to analysis of stability and dwell time of a certain class of switched systems. The studied class represents the systems switching beetween two subsystems that have Hurwitz matrices of the second order. The first of those matrices is presented in the Frobenius canonical form, and the second one has the same eigenvalues, but its components of the state vector are ``swapped'' and it has the corresponding realization. The considered problem is to find all eigenvalues which provide asymptotic stability of the switched system under an arbitrary switching signal and any non-zero initial conditions. In the case of eigenvalues, which provide no asymptotic stability under arbitrary switching signal, the problem is to find an estimation of the system dwell time that guarantees it. The study uses known approaches for asymptotic stability analysis and for dwell time estimation.