I read this [post](https://math.stackexchange.com/questions/994847/adjacency-matrix-and-connectivity-proof?rq=1) about graph connectedness and adjacency matrix. But I do not understand how the last comment about using the binomial expansion would help. I know this approach is correct, but just do not fully grasp it. 

Would you mind to elaborate a little bit more? I did the binomial expansion and got $\binom{n-1}{0}A^{0}+\binom{n-1}{1}A^{1}+.....+\binom{n-1}{n-1}A^{n-1}$ which is a linear combination of all the As. but I still don't know how this helps.