<s>[INST] 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? 
 [/INST] Hmm, you can also prove it from basic understanding of inverses of triangular matrices, and a proof by induction.
$$ \left( \begin{array}{c | c}
A_{TL} & A_{TR} \\ \hline
0 & A_{BR}
\end{array} \right)^{-1} =
\left( \begin{array}{c | c}
A_{TL}^{-1} & -A_{TL}^{-1} A_{TR} A_{BR}^{-1}\\ \hline
0 & A_{BR}^{-1}
\end{array} \right)
$$
Now, let's use $ A_{n \times n} $ to equal a matrix of the given form, of size $ n \times n $,
and $ x_n = \left( \begin{array}{c}
1 \\
-1 \\
1 \\
\vdots
\end{array} \right)
$, of size $ n $.
Notice that
$$ A_{n \times n} = 
\left( \begin{array}{c | c}
A_{k \times k} & (-1)^k x_k  x_{(n-k)}^T \\ \hline
0 & A_{(n-k) \times (n-k)}
\end{array} \right)
$$
where
$$
x_k x_{n-k}^T = \left( \begin{array}{c}
1 \\
-1 \\
1 \\
\vdots
\end{array} \right)
\left( \begin{array}{c}
1 \\
-1 \\
1 \\
\vdots
\end{array} \right)^T
=
\left( \begin{array}{c}
1 \\
-1 \\
1 \\
\vdots
\end{array} \right)
\left( \begin{array}{c c c c}
1 & -1 & 1 & \cdots
\end{array}
\right)
=
\left( \begin{array}{c c c c}
1 & -1 & 1 & \cdots \\
-1 & 1 & -1 & \cdots \\
1 & -1 & 1 & \cdots \\
\vdots & \vdots & \vdots & 
\end{array} \right)
$$
So...
$$ A_{n \times n}^{-1} =
\left( \begin{array}{c | c}
A_{k \times k}^{-1} & - (-1)^k A_{k \times k}^{-1} x_k  x_{(n-k)}^T  A_{(n-k) \times (n-k)}^{-1} \\ \hline
0 & A_{(n-k) \times (n-k)}^{-1}
\end{array} \right)
$$
Now, 
$$
A_{k \times k}^{-1} x_k = 
-(-1)^{k} e_L
$$
where $ e_L $ is a vector of all zeroes except the last entry, which equals one.
Similarly,
$$
x_{(n-k)}^T A_{k \times k}^{-1} = 
e_F
$$
where $ e_F $ is a vector of all zeroes except the first entry, which equals one.
Now, I am notorious for loosing negative signs and minor typos, but you get the idea.
</s>