Given a 4 by 4 matrix, $A$, as below, find its inverse matrix, $A^{-1}$.

The sum of all the elements of $A^{-1}$ is $b$. Rounding $|b|$ to the nearest integer, $c$. What is the value of mod($c$,5), where mod(.) is the modulo operation. (For example, mod(5,2)=1.)