dsj comments on Whence the determinant?

[dsj]

det(AB) = det(A)det(B), so the determinant is invariant to any change of basis, not merely unitary ones:

$det\left(AB{A}^{-1}\right)$ $=det\left(A\right)det\left(B\right)det\left(A^{-1}\right)$ $=det\left(A{A}^{-1}\right)det\left(B\right)$ $=det\left(B\right)$