How many zero elements are there in the inverse of the $n\times n$ matrix
$A=\begin{bmatrix} 1&1&1&1&\cdots&1\\ 1&2&2&2&\cdots&2\\ 1&2&1&1&\cdots&1\\ 1&2&1&2&\cdots&2\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ 1&2&1&2&\cdots&\cdots \end{bmatrix}$
My try: Denote by $a_{ij}$ and $b_{ij}$ the elements of $A$ and $A^{-1}$,respectively then for $k\neq m$,we have $$\sum_{i=0}^{n}a_{ki}b_{im}=0$$
Then I can't.Thank you very much
Investigation with Maple indicates that for $n>1$, the inverse of $A$ has the following form:
(1) the main diagonal starts with $2$, finishes with $\pm1$, has zeros in between;
(2) the next diagonal each way has $1$ and $-1$ alternating;
(3) everything more than "one step away" from the main diagonal is $0$.
For example $$\pmatrix{1&1&1&1\cr 1&2&2&2\cr 1&2&1&1\cr 1&2&1&2\cr}^{-1} =\pmatrix{2&-1&0&0\cr -1&0&1&0\cr 0&1&0&-1\cr 0&0&-1&1\cr}$$ and $$\pmatrix{1&1&1&1&1\cr 1&2&2&2&2\cr 1&2&1&1&1\cr 1&2&1&2&2\cr 1&2&1&2&1\cr}^{-1} =\pmatrix{2&-1&0&0&0\cr -1&0&1&0&0\cr 0&1&0&-1&0\cr 0&0&-1&0&1\cr 0&0&0&1&-1\cr}\ .$$ I can't see a simple proof that this is true for all $n$, but if it is then the number of zeros is $n^2-2n$, provided $n>1$.
I am excavating this post because this post needs an affirmative answer.
After some try I feel that this can be (ironically) done fastest by brute force. Let $A^{-1} = (a_{ij})$ and we just solve column-wisely $AA^{-1} = I$.
For the first column, by subtracting equations derived from neighboring row of $A$, you will get:
$$ \begin{aligned} a_{11} + a_{21} + a_{31} + a_{41} + \cdots &= 1\\ a_{21} + a_{31} + a_{41} + \cdots &= -1 \\ -a_{31}- a_{41} - \cdots &= 0 \\ a_{41} + \cdots &= 0\\ &\vdots \end{aligned} $$
You then have $a_{11} = 1$, $a_{21} = -1$ and the rest are 0.
From the second to the penultimate columns, $a_{ij}$ for $i < j$ can be derived directly from the fact that $A^{-1}$ is symmetric as $A$ is, for the rest, play the same "compare neighboring row game", you will get:
$$ \begin{aligned} (-1)^ia_{ii} + (-1)^ia_{i+1,i} + (-1)^ia_{i+2,i} + \cdots &= 1\\ (-1)^{i+1}a_{i+1,i} + (-1)^{i+1}a_{i+2,i} + \cdots &= -1 \\ (-1)^{i+2}a_{i+2,i} \cdots &= 0 \\ &\vdots \end{aligned} $$
As the equations are easy enough, you will immediately see that $a_{i, i+1} = (-1)^i$ and $a_{i, i-1} = (-1)^{i-1}$.
For the last column, the only thing new is that $(-1)^na_{nn} = 1$, so $a_{nn} = (-1)^n$ and $a_{n, n-1} = (-1)^{n-1}$.
At the end you see that $A^{-1}$ has $2n$ non-zero entries, where exactly 2 of them are on each row and each column.
This isn't a complete answer, either, but note that your matrix can be written as
$$\pmatrix{ 1\cr \vdots \cr 1} \pmatrix{1 & \cdots & 1} + \pmatrix{ 0 \cr 1 \cr \vdots \cr 1} \pmatrix{0 & 1 &\cdots & 1 } - \pmatrix{ 0 \cr 0 \cr 1 \cr \vdots \cr 1}\pmatrix{ 0 & 0 & 1 & \cdots & 1}+\ldots$$
Because the matrix is symmetric all its eigenvalues are real and all its eigenvectors can be orthogonalized.
If the bottom right element equals 2 then you get something like
$$\pmatrix{0 \cr\vdots \cr 0 \cr 1} \pmatrix{0 &\cdots & 0 & 1} + \pmatrix{0 \cr \vdots \cr 0 \cr 1 \cr 1 \cr 0}\pmatrix{0 & \cdots & 0 & 1 & 1 & 0} + \dots$$
for the positive eigenvalues and something similar for the negative eigenvalues.
Hth
