Consider a multi-diagonal matrix with invariant factors along the diagonals : $$ \left[ \begin{matrix} a_1 & a_2 & a_3 & 0 & \dots & \dots & \dots & \dots & 0 \\ a_2 & a_1 & a_2 & a_3 & 0 & \dots & \dots& \dots & 0\\ a_3 & a_2 & a_1 & a_2 & a_3 & 0 & \dots &\dots& 0\\ 0 & a_3 & a_2 & a_1 & a_2 &a_3 & 0& & 0\\ \vdots & \ddots & \ddots & \ddots & \ddots &\ddots &\ddots &\ddots& \vdots \\ 0 & \dots & 0& a_3& a_2& a_1 &a_2&a_3 & 0\\ 0 & \dots & \dots& 0& a_3& a_2 &a_1&a_2&a_3\\ 0 & \dots & \dots& \dots& 0& a_3 &a_2&a_1&a_2\\ 0 & \dots & \dots& \dots& \dots& 0 &a_3&a_2&a_1 \end{matrix} \right] $$
Are there some nice properties about the inverse of this matrix ? Can we say that the inverse has invariant factors along the diagonal ? Is the inverse formulation independent of the rank of the matrix ?