Decomposing a matrix into a product of sparse matrices How to study the decomposition of a square matrix into a product of sparse matrices? 
There are no restrictions on the number of matrices in the product, but the fewer the better.
 A: The question of deciding if a specific matrix is a product of "a few" "sparse" matrices seems very interesting to me and perhaps rather hard. 
Likewise the question of how bad things could be if the underlying field is finite (say $\mathbb{Z}_2.$)
However a random matrix is, in some sense, highly unlikely to have any unusual decompositions.
Here is a rather extreme conjecture which seems plausible to me: define the density of an $ n \times n$ matrix to be the number entries which are not $0$ or $1$ and the density of a product of such matrices to be the sum of the densities of the factors. Then with probability $1$ a random real matrix has density $n^2$ and, more generally, no factorization has lower density. However every matrix is a product of $n^2$ matrices of density $1$ and , with probability $1$, there are $n^2!$ such decompositions.
Perhaps a better definition , at least for integer matrices (with huge entries), would be the number of entries in which a matrix differs from the identity matrix or the previous definition plus $1.$
