  In comparing geodesics induced by different metrics, Audenaert formulated the
following determinantal inequality
  $$\det(A^2+|BA|)\le \det(A^2+AB),$$ where $A, B$ are $n\times n$ positive
semidefinite matrices. We complement his result by proving
  $$\det(A^2+|AB|)\ge \det(A^2+AB).$$ Our proofs feature the fruitful interplay
between determinantal inequalities and majorization relations. Some related
questions are mentioned.
