Showing determinant of product is product of dets for linear operators

[Hakkinen]

Homework Statement

Assume A and B are normal linear operators $[A,A^{t}]=0$ (where A^t is the adjoint)

show that det AB = detAdetB

Homework Equations

The Attempt at a Solution

Well I know that since the operators commute with their adjoint the eigenbases form orthonormal sets. I'm not really sure how to proceed though so any assistance would be appreciated.

 

[Hakkinen]

I think I've got a solution for this but I'm not sure if it's general enough.

So since each operators eigenbasis forms an orthonormal set you can choose any convenient orthonormal basis. If I pick the standard cartesian basis (i,j,k) then each operator can be represented by the identity matrix. From here it is straight forward to show the identity.