Regular polynomial matrices having relatively prime determinants

It is shown that a necessary and sufficient condition that two regular polynomial matrices T, U have relatively prime determinants is that the equation TX + YU = E , where E is a constant matrix, has a unique solution with the degrees of X, Y less than the degrees of U, T respectively. This is a generalization of a well-known theorem for scalar polynomials. An alternative form for the condition in terms of the non-vanishing of a determinant, corresponding to the resultant of two scalar polynomials, is also obtained together with an equivalent determinant of lower order.