Question: At the end of this chapter, the claim was made that "even very sparse matrices tend to have dense inverses." Artificially constructed examples do not support empirical claims such as this one very convincingly, but here is one anyway:
Compute a triangular factorization of A and compare its density with the density of A-I. Generalize this example to arbitrary sizes n x n such that n is odd. (Why is there no generalization with n even?)