  We investigate the relation between the spectrum of matrix (or operator)
polynomials and the Taylor spectrum of its coefficients. We prove that the
polynomial of commuting matrices is singular, i.e. its spectrum is the whole
complex plane, if and only if (0, 0, ... , 0) belongs to the Taylor spectrum of
its coefficients. On the other hand we prove that this equivalence is not
longer true if we consider the operators on infinite dimensional Hilbert space
as coefficients of polynomial. As a consequence we could propose a new
description of (Taylor) spectrum of k-tuple of matrices and we could disprove
the conjecture previously proposed in the literature. Additionally, we pointed
out the Kronecker forms of the pencils with commuting coefficients.
