# Determinant where the matrix entries may not be commutative

I am trying to find definition of determinant where the matrix entries may not be commutative, for instance quaternions or Clifford algebras. Does the minor expansion of determinant over a field still work over a non-commutative algebra?

JasonRox
Homework Helper
Gold Member
I am trying to find definition of determinant where the matrix entries may not be commutative, for instance quaternions or Clifford algebras. Does the minor expansion of determinant over a field still work over a non-commutative algebra?

It doesn't seem like the definition assume commutativity. So, I guess it would work.

Hurkyl
Staff Emeritus
Gold Member
A google search on "noncommutative determinant" turns up several hits -- maybe one will have a useful definition?

Hurkyl
Staff Emeritus
Gold Member
My gut says that a useful definition of determinant could be defined as follows:

Your algebra A is a finite dimensional algebra over the reals -- I think that implies that it can be represented as an algebra of real matrices. So you can view your matrix over A as a matrix over R, and take the (real-valued!) determinant of that.

My gut says that a useful definition of determinant could be defined as follows:

Your algebra A is a finite dimensional algebra over the reals -- I think that implies that it can be represented as an algebra of real matrices. So you can view your matrix over A as a matrix over R, and take the (real-valued!) determinant of that.

However, that is not that I am working with. I am playing with Clifford algebra and it is known that every element of Cl(p+1, q+1) can be represented by M(2, Cl(p,q)) and M(n, T) where T = R, C or H. Of course the definition of determinant is trivial when T = R or C, but I am not sure if the Laplace expansion still works on quaternionic matrices. If I can compute the determinant in that case, it will suggest the norm function for Cl(p+1, q+1).

Thank you for your suggestion by the way. I searched with the keyword pseudodeterminant while the suggested keyword 'noncommutative determinant' is more obvious. 