# Showing determinant of product is product of dets for linear operators

1. Nov 25, 2013

### Hakkinen

1. The problem statement, all variables and given/known data
Assume A and B are normal linear operators $[A,A^{t}]=0$ (where A^t is the adjoint)

show that det AB = detAdetB

2. Relevant equations

3. 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.

2. Nov 25, 2013

### 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.