Eigenvalues · Superchallenge problem

Let $T$ be the linear map on the $9$-dimensional space of real $3\times 3$ matrices given by $T(X) = X^{T} + (\operatorname{tr}X)\,I$. Find $\det(T + 2\,\mathrm{id})$, where $\mathrm{id}$ is the identity map.