Matrices & Systems · Superchallenge problem

Let $A$ be the $4\times 4$ matrix with entries $a_{ij} = \dfrac{1}{i + j}$ for $1 \le i, j \le 4$ (so its first row is $\tfrac12, \tfrac13, \tfrac14, \tfrac15$). $A$ is invertible. Find the sum of all $16$ entries of $A^{-1}$.