In this note we use Floquet theory to establish that a sufficient condition for the existence of a periodic destabilizing switching sequence for the system x = A(t)x, A(t) is an element of {A(1), A(2),..., A(M)}, Ai is an element of R-NxN, where the A(i) are Hurwitz matrices, is given by the existence of non-negative real constants alpha(1), alpha(2),...,alpha(M), alpha(i) >= 0, Sigma(M)(i=1) alpha(i) > 0, such that the matrix pencil Sigma(M)(i=1) alpha(i)A(i) has at least one eigenvalue with a positive real part. We use this result to provide insights into the relationship between the non-existence of a common quadratic Lyapunov function and the existence of a destabilizing switching sequence for low order systems, as well as the robustness of a class of switching system that is known to be exponentially stable.