Answer
-8
Work Step by Step
If A and B are $4×4$ matrices, then:
$\det(AB)=\det(A).\det(B)$
Also $\det(A^T)=\det(A)$
and if $\det(A) \ne 0$ then $det(A^{−1})=\frac{1}{\det(A)}$
Hence here,
$\det(((A^{−1}B)^T)(2B^{−1}))$
$=\det((A^{−1}B)^T)\det(2B^{−1}))$
$=\det(A^{-1}B)2^4\det(B^{-1})$
$=16\det(A^{-1}).\det(B)\det(B^{-1})$
$=1\frac{1}{\det(A)}\det (B). \frac{1}{\det (B)}$
$=\frac{16}{\det (A)}=\frac{16}{-2}=-8$
