We study under which conditions the maximal-rank minors of a (possibly singular) Macaulay matrix vanish. The Macaulay matrix is a matrix whose entries are the coefficients of a system of multivariate polynomial equations. The determinant of the Macaulay matrix vanishes if the polynomial system has a common root. Macaulay matrices have applications in many areas of computing, such as computer aided geometric design, robotics, computational chemistry, etc. It is shown that the vanishing of the maximal-rank minors of the Macaulay matrix of a parametric system of polynomials under specialization is a necessary condition for the specialized polynomials to have an additional common root even when the parametric system has common roots without any specialization of parameters. This result has applications where conditions for additional common roots of polynomial systems with generic roots are needed, such as in implicitization of surfaces with base points and in various other areas of computational geometry. We will discuss such applications.