  We prove pinching estimates for solutions of the linearized Ricci flow system
on a closed manifold of dimension $n\geq 4$ with positive scalar curvature and
vanishing Weyl tensor. If the vanishing Weyl tensor condition is removed, we
only give a rough pinching estimate controlled by some blow-up function in a
short time. These results extend the $3$-dimensional case due to Anderson and
Chow (2005) \cite{[AC]}.
