Recognizing Patterns in Alternating Signs Matricies and their Inverses Let's say we have the matrix A with alternating-sign 1's below
A = \begin{bmatrix}1&-1&1&-1\\0&1&-1&1\\0&0&1&-1\\0&0&0&1\end{bmatrix}
If we find the inverse, we get 
A^-1 = \begin{bmatrix}1&1&0&0\\0&1&1&0\\0&0&1&1\\0&0&0&1\end{bmatrix}
We get a similar pattern for 5 x 5, 6 x 6, ..., n x n matricies. 
How would we prove that we would achieve this pattern for all inverses of n x n matricies? 

See https://en.wikipedia.org/wiki/Matrix_function   and the reference Higham (2008)
Also https://en.wikipedia.org/wiki/Jordan_normal_form#Matrix_functions
The matrix you call $A^{-1}$ is already in Jordan normal form. It is already written as $I + N,$ where $I$ is the identity and $N$ is the set of $1's$ off the diagonal. The importance of $N$ as a matrix is that it is nilpotent, in particular $N^n = 0$
So, we have the series
$\frac{1}{1 + x} = 1 - x + x^2 - x^3 + x^4 - x^5 ...$
which converges when $-1 < x < 1.$
In comparison, the fact that $N$ is nilpotent, and $IN=NI,$ means
$$ (I+N)^{-1} = I - N + N^2 - N^3 + N^4 - + \cdots + (-1)^{(n-1)} N^{n-1} $$
and there is no need to worry about convergence. 
This is a basic fact of life. Given a real analytic function, we can evaluate that function on a square matrix if we know how to find the Jordan normal form of the matrix. There are whole books on this.
