Singular Values & the SVD · Challenge problem

Let $A = \begin{pmatrix}1 & 2 & 2\\4 & 2 & -4\end{pmatrix}$ and let $H = \begin{pmatrix}0 & A\\A^{T} & 0\end{pmatrix}$, a symmetric $5\times 5$ matrix. Find $\det(I_5 + H)$.