In scipy.sparse.linalg package there is a solver called spsolve that solves
$$ AX = b $$
where $A,b$ are a sparse matrices.
Here there is a claim being made that the result $X$ is often sparse, when $A$ and $b$ are very sparse.
However, from my understanding and experiments I did, the inverse of a sparse matrix $A$ is rarely ever sparse, and the product of a dense matrix with a sparse matrix may never be sparse. So how can we make the claim that $X$ is usually sparse?
My second question is concerning the solver used in spsolve. There is no literature reference on their webpage, so I am wondering if anyone knows the type of solver scipy uses, which would be helpful to understand under what conditions the result is sparse