We give an alternative, more convenient expression of the well known Cayley-Hamilton theorem when polynomial matrices of arbitrary degree are involved. Based on the results of a recently developed algorithm (Fragulis et al. 1991) for the computation of the inverse of a polynomial matrix, certain relationships among the coefficient matrices of the given polynomial matrix are obtained. We also propose two ways of finding the powers of a polynomial matrix : one in terms of its coefficient matrices and the other making use of the generalized Cayley-Hamilton theorem. These methods are of closed form and are easily implementable in a digital computer.