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While putzing around with the linear algebra capabilities of my computing environment, I noticed that inverses of $n\times n$ matrices $\mathbf M$ associated with a sequence $a_i$, $i=1\dots n$ with $m_{ij}=a_{\max(i,j)}$, which take the form

$$\mathbf M=\begin{pmatrix}a_1&a_2&\cdots&a_n\\a_2&a_2&\cdots&a_n\\\vdots&\vdots&\ddots&a_n\\a_n&a_n&a_n&a_n\end{pmatrix}$$

(i.e., constant along "backwards L" sections of the matrix) are tridiagonal. (I have no idea if there's a special name for these matrices, so if they've already been studied in the literature, I'd love to hear about references.) How can I prove that the inverses of these special matrices are indeed tridiagonal?

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