# Inverse of tridiagonal Toeplitz matrix has no zero entries

The inverse of the symmetric tridiagonal matrix (Toeplitz) t_{ij}=\begin{align} \begin{cases} -2 &\quad \text{if} \,\, i=j \\ 1 &\quad\text{if} \,\, \vert i-j\vert = 1 \end{cases} \end{align}

does not have any non zero entries according to online inverse calculators (I tried up to $$5 \times 5-$$matrices). Why is it so?

The expression for the $$i,j$$ element of the inverse of this $$n\times n$$ matrix has a nice form \begin{align} w_{ij}&=\frac{i\,j}{n+1}-\min(i,\,j) \tag{1}\label{1} , \end{align}
so $$w_{ij}$$ could be zero only if \begin{align} n&=\frac{i\,j}{\min(i,\,j)}-1 =\max(i,j)-1=n-1 , \end{align}
• How did you obtain the expression for the $i,j$ element of the inverse? just looking at it and recognizing the pattern? Nov 10, 2019 at 8:49