$\mathrm{A}=\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\end{array}\right]\mathrm{B}=\left[\begin{array}{l}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\end{array}\right]$

${\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}=\left[\begin{array}{l}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\end{array}\right]\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\end{array}\right]=\left[\begin{array}{l}2+6\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}6+12\\ 1+8\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3+16\end{array}\right]=\left[\begin{array}{l}8\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}18\\ 9\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}19\end{array}\right]$

det(B^TA^T) = 8 × 19 – 9 × 18 = 152 – 162 = – 10

${\left({\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\right)}^{-1}=\frac{1}{\det \left({\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\right)}\mathrm{adj}\left({\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\right)=\frac{1}{-10}\left[\begin{array}{cc}19& -18\\ -9& 8\end{array}\right]$

$\det \left({\left({\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\right)}^{-1}\right)=\frac{19\times 8-18\times 9}{-10}=1$