<s>[INST] Finding an inverse matrix of an infinite matrix I tried to find the inverse matrix of 
$$
        \begin{pmatrix}
        2 & -1 & 0 & \cdots& 0   \\
        -1 & 2 & -1 & \cdots& 0\\
        0 & -1 & 2 & \ddots& \vdots\\
        \vdots & \vdots & \ddots & \ddots &-1\\
        0 & 0  & \cdots & -1& 2\\
        \end{pmatrix}
$$
unsuccesfuly. 


*

*For $n=2$ the inverse matrix is $$\frac 1 3
        \begin{pmatrix}
        2 & 1 \\
        1 & 2 \\
        \end{pmatrix}
$$

*For $n=3$ the inverse matrix is 
$$
        \frac 1 4\begin{pmatrix}
        3 & 2 & 1 \\
        2 & 4 & 2 \\
        1 & 2 & 3 \\
        \end{pmatrix}
$$


*

*In general I figured out:


$$
        \frac 1 {n+1} \begin{pmatrix}
        n & n-1 & n-2 & \cdots& 1   \\
        n-1 & ? & ? & ?& 2\\
        n-2 & ? & ? & ? & \vdots\\
        \vdots &?  & ? & ? &n-1\\
        1 & 2  & \cdots & n-1& n\\
        \end{pmatrix}
$$
I just don't know how to write it generally even though I see those numbers in the middle of my matrices for $n = 2$ and $n = 3$. I tried it even for the larger $n$. 
 [/INST] Your examples suggest that the $(i,j)$-th entry in the upper triangular part (i.e. when $i\le j$) is $\frac{1}{n+1}i(n+1-j)$. It should be straightforward to verify this formula.
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