I have a decent understanding of the determinant of a matrix in terms of its role in
- Telling you if a matrix is invertible (zero vs. nonzero)
- Expressing the product of a matrix's eigenvalues with multiplicities
- Representing the constant term in a matrix's characteristic polynomial
- Having geometrical interpretations
However, I was curious to learn in the operation known as the permanent has any interesting properties. The permanent is defined in the same way a determinant is, but all entries are added instead of alternating between positive and negative terms.
In particular, what are the significant applications of defining the permanent as a notable matrix operation? Also, are there any matrix problems whose solutions directly/indirectly depend on understanding the permanent?