  We prove that given a solution of the Einstein equations $g_{ab}$ for the
matter field $T_{ab}$, an autoparallel null vector field $l^{a}$ and a solution
$(l_{a}l_{c}, \mathcal{T}_{ac})$ of the linearized Einstein equation on the
given background, the Kerr-Schild metric $g_{ac}+\lambda l_{a}l_{c}$ ($\lambda
$ arbitrary constant) is an exact solution of the Einstein equation for the
energy-momentum tensor $T_{ac}+\lambda \mathcal{T}_{ac}+\lambda
^{2}l_{(a}\mathcal{T}_{c)b}l^{b}$. The mixed form of the Einstein equation for
Kerr-Schild metrics with autoparallel null congruence is also linear. Some more
technical conditions hold when the null congruence is not autoparallel. These
results generalize previous theorems for vacuum due to Xanthopoulos and for
flat seed space-time due to G\"{u}rses and G\"{u}rsey.
