  For any quiver mutation sequence, we define a pair of matrices that describe
a fixed point equation of a cluster transformation determined from the mutation
sequence. We give an explicit relationship between this pair of matrices and
the Jacobian matrix of the cluster transformation. Furthermore, we show that
this relationship reduces to a relationship between the pair of matrices and
the $C$-matrix of the cluster transformation in a certain limit of cluster
variables. As an application, we prove that quivers associated with
once-punctured surfaces do not have maximal green or reddening sequences.
