Define a sparse matrix set where a matrix and its inverse have the same sparse structure. Is such a set forming a certain group?
For example, $\begin{bmatrix}a & b\\0 & c \end{bmatrix}^{-1}=\begin{bmatrix}1/a & -b/(ac)\\0 & 1/c \end{bmatrix}$ has the same sparse structure with $\begin{bmatrix}a & b\\0 & c \end{bmatrix}$. It belongs to the set defined above.