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Two sufficient conditions under which the roots of the determinant of a given ( ) matrix polynomial of th order lie in the open left-half plane have been obtained. The first condition is given in terms of the positive definiteness of an ( ) symmetric matrix, while the second condition is given in terms of the positive definiteness of an ( ) matrix that is a function of , Re . These conditions are represented in terms of rational functions of the coefficient matrices of the given matrix polynomial. Therefore, the explicit computation of the determinant polynomial is not required.