Use multilinearity of determinant:
$$\begin{vmatrix}na_1+b_1&na_2+b_2&na_3+b_3\\
nb_1+c_1&nb_2+c_2&nb_3+c_3\\
nc_1+a_1&nc_2+a_2&nc_3+a_3\end{vmatrix}=\begin{vmatrix}na_1&na_2&na_3\\
nb_1&nb_2&nb_3\\
nc_1&nc_2&nc_3\end{vmatrix}+\begin{vmatrix}na_1&na_2&b_3\\
nb_1&nb_2&c_3\\
nc_1&nc_2&a_3\end{vmatrix}+$$
$$+\begin{vmatrix}na_1&b_2&na_3\\
nb_1&c_2&nb_3\\
nc_1&a_2&nc_3\end{vmatrix}+\begin{vmatrix}na_1&b_2&b_3\\
nb_1&c_2&c_3\\
nc_1&a_2&a_3\end{vmatrix}+\begin{vmatrix}b_1&na_2&na_3\\
c_1&nb_2&nb_3\\
a_1&nc_2&nc_3\end{vmatrix}+\ldots$$
Observe that if we put
$$\Delta=\begin{vmatrix}a_1&a_2&a_3\\
b_1&b_2&b_3\\
c_1&c_2&c_3\end{vmatrix}$$
then we have that the four first determinants above equal (factor out constants from rows/columns):
$$n^3\Delta+n^2\Delta+n^2\begin{vmatrix}a_1&b_2&a_3\\b_1&c_2&b_3\\c_1&a_2&c_3\end{vmatrix}+n\begin{vmatrix}a_1&b_2&b_3\\b_1&c_2&c_3\\c_1&a_2&a_3\end{vmatrix}+\ldots$$
Well, develop the other three determinants left and sum up all.