A sufficient condition for the existence of a destabilising switching sequence for the system x = A(t)x, A(t) E {Al,A2 , ... ,AM}, Ai E lRNXN , where the Ai are Hurwitz matrices, is that there exists non-negative real constants 0'1,0'2, ... , O'M, O'j 2: 0, L:f'!1 0'; > 0, such that the matrix pencil L:f'! IO'jAj has at least one eigenvalue with a positive real part. An informal proof of this result based upon Floquet theory was presented in (Shorten, 1996; Shorten and Narendra, 1997) . In this paper we present a rigourous basis for the proof of this result. Further, we use this result to identify several classes of linear switching systems, which admit the existence of a destabilising switching sequence. These systems provide insights into the relationship between the existence of a common quadratic Lyapunov function and the existence of a destabilising switching sequence for low order systems, as well as the robustness of a class of switching system that is known to be exponentially stable.