  Let $\Omega\subset\er^3$ be a domain and let $f\colon\Omega\to\er^3$ be a
bi-$BV$ homeomorphism. Very recently in \cite{HKL} it was shown that the
distributional adjugate of $Df$ (and thus also of $Df^{-1}$) is a matrix-valued
measure. In the present paper we show that the components of $\Adj Df$ are
equal to components of $Df^{-1}(f(U))$ as measures and that the absolutely
continuous part of the distributional adjugate $\Adj Df$ equals to the
pointwise adjugate $\adj Df(x)$ a.e. We also show the equivalence of several
approaches to the definition of the distributional adjugate.
