Proof: Sum of weights of paths in a network
A few hints:


Find out what the entries of $A^2$ mean. How about $A^k$? (Start by a very small adjacency matrix, multiply it with itself and try to think what each entry means).
Lookup and understand the Taylor expansion of $f(x) = 1/(1-x)$.
Realize that the sum of weights of all paths from $s$ to $t$ is the sum of weights of paths of length 0, 1, 2, ...