Eigenvalues · Superchallenge problem

Let $A = \begin{pmatrix}1 & 0 & 1\\0 & 1 & 1\end{pmatrix}$ and $B = \begin{pmatrix}1 & 2\\0 & 1\\1 & 0\end{pmatrix}$. Find $\det(I_3 + BA)$, where $I_3$ is the $3\times3$ identity. (Hint: compare the eigenvalues of $BA$ and $AB$.)