Diagonalization & Matrix Powers · Superchallenge problem

Let $A = \begin{pmatrix}1 & 2 & 0\\0 & 3 & 1\\0 & 0 & 6\end{pmatrix}$ and define the linear map $T$ on the $9$-dimensional space of real $3\times 3$ matrices by $T(X) = AX - XA$. Find $\det(T + \mathrm{id})$, where $\mathrm{id}$ is the identity map.