In this paper, a definition of a class of rational function matrices called a type-1 matrix is given. Two illustrative examples show that the type-1 matrix can describe most of linear physical systems. It is proven that the type-1 matrix satisfies two properties that its characteristic polynomial has no nonzero constant eigenvalues or nonzero multiple factors. The author presents some controllability criteria of the linear systems whose characteristic polynomials have no nonzero multiple factors. The applications of the type-1 matrix and system to structural controllability are indicated.