We have a special matrix and we would like to find close form or analytical formula for its inverse. The matrix is symmetric and it has only first and last row elements, first and last column elements and main diagonal. I am wondering if there is a name for this matrix and how we can find the inverse of it?
\begin{align} \begin{split} \mathbf{A} =\begin{bmatrix} a_{11}&a_{12}&a_{13}&a_{14}&...&a_{1n}\\% a_{12}&a_{22}&0&0&...&a_{2n}\\% a_{13}&0&a_{33}&0&...&a_{3n}\\% a_{14}&0&0&a_{44}&...&a_{4n}\\% ...&...&...&...&...&...\\% a_{1n}&a_{2n}&a_{3n}&a_{4n}&...&a_{nn}\\% \end{bmatrix} \end{split} \end{align} This matrix can can also be considered as a sum of a diagonal matrix and a upper triangular matrix and its transpose : \begin{align} \begin{split} \mathbf{A}=\mathbf{labmda}+\mathbf{A_{u}}+\mathbf{A_{u}}^{T} \end{split} \end{align} where
\begin{align} \begin{split} \mathbf{labmda} =\begin{bmatrix} a_{11}&0&0&0&...&0\\% 0&a_{22}&0&0&...&0\\% 0&0&a_{33}&0&...&0\\% 0&0&0&a_{44}&...&0\\% ...&...&...&...&...&...\\% 0&0&0&0&...&a_{nn}\\% \end{bmatrix} \end{split} \end{align}
\begin{align} \begin{split} \mathbf{A_{u}} =\begin{bmatrix} 0&a_{12}&a_{13}&a_{14}&...&a_{1n}\\% 0&0&0&0&...&a_{2n}\\% 0&0&0&0&...&a_{3n}\\% 0&0&0&0&...&a_{4n}\\% ...&...&...&...&...&...\\% 0&0&0&0&...&0\\% \end{bmatrix} \end{split} \end{align}