I have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices).

I encountered several times the web pages which states that the inverse of the sparse matrix is not usually sparse and my experience also said so. One exception can be diagonal matrices.

How theses kind of assertions can be verified?

EDIT: As suggested by Yemon Choi, it would be better to ask if there is a precise way to say that the inverses of "most" sparse matrices are not sparse.

NOTE : This was asked in math stack exchange earlier, but no one seems to answer it so I posted here too. Looking forward to great answers.:)