This paper investigates the problem of designing a linear state feedback control to stabilize a new class of single-input uncertain linear dynamical systems. Uncertain parameters in the system matrices are time-varying and bounded in given compact sets. We first provide a concept called "new standard system", where some of the entries are required to be negative sign-invariant and sign-invariant, and each entry varies independently in an arbitrarily large range. Then, for a class of new standard systems we derive a necessary and sufficient condition under which a system can be quadratically stabilized by a linear control for all admissible variations of uncertainties. The result extends the main result in Wei (1990. IEEE Transactions on Automatic Control, 35(3), 268-277).