Determinants & Inverses · Superchallenge problem

Let $A = \begin{pmatrix}1 & 0\\1 & 1\\0 & 1\\2 & 0\\1 & 2\end{pmatrix}$ (a $5\times2$ matrix) and $B = \begin{pmatrix}1 & 1 & 0 & 0 & 1\\0 & 1 & 1 & 1 & 0\end{pmatrix}$ (a $2\times5$ matrix). Compute $\det(I_5 + AB)$.