Determinants & Inverses · Challenge problem

Let $A$ be the $4\times4$ matrix with entries $a_{ij} = 2^{|i-j|}$, that is, $A = \begin{pmatrix}1 & 2 & 4 & 8\\2 & 1 & 2 & 4\\4 & 2 & 1 & 2\\8 & 4 & 2 & 1\end{pmatrix}$. Compute $\det A$.