# What are some Applications of the Permanent of a Matrix?

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?

• math.stackexchange.com/questions/761297/… – Moishe Kohan Jan 5 '15 at 2:44
• @studiosus I like the information in your link from a theoretical standpoint, but it doesn't quite answer the question. I'm looking for a more application-oriented answer or problem that can be solved using permanents, not necessarily knowing about the time complexity to compute them. – Xoque55 Jan 5 '15 at 2:50
• Well, if you really understand permanents, you will prove that P equals NP or, vice versa, that P is not equal to NP; this will have major ramifications in the computer science and will earn you $10^6$ USD among other things. I guess, however, this is not an answer you are looking for, therefore I posted this link just as a comment. (+1 for the question itself.) – Moishe Kohan Jan 5 '15 at 4:26