A Generalization of the Cayley-Hamilton Theorem

Let A = [a(ij)](nxn) and B = [b(ij)](nxn) be two commuting square matrices of order n over an arbitrary commutative ring. We prove that the determinant of the matrix [b(ij)A - a(ij)B](nxn) which is regarded as an n x n block matrix with pairwise commuting entries, is exactly equal to the n x n zero matrix. If B is the identity matrix, then the result is equivalent to the Cayley-Hamilton theorem.