Let A and B are 3$\times$3 real matrices which are non – singular. Given that $\mathrm{A}=\mathrm{adjB}-{\mathrm{B}}^{\mathrm{T}} \mathrm{and} \mathrm{B}=\mathrm{adjA}-{\mathrm{A}}^{\mathrm{T}}$ and sum of digits in the value of the expression ${\left(\det \mathrm{A}\right)}^2+{\left(\det \mathrm{B}\right)}^2 \mathrm{equals} \mathrm{K} \mathrm{then}\left[\frac{\mathrm{K}}{2}\right] \mathrm{equals} \left(\left[.\right] \mathrm{is} \mathrm{GIF}\right)$

$$\begin{gathered} \mathrm{A}=\mathrm{adjB}-{\mathrm{B}}^{\mathrm{T}} \Rightarrow A^T={\left(\mathrm{adj}\left(B\right)\right)}^T-B \\ \begin{array}{r} \mathrm{B}=\mathrm{adjA}-{\mathrm{A}}^{\mathrm{T}}\\ B=\mathrm{adj}\left(A\right)-{\left(\mathrm{adj}\left(B\right)\right)}^T+B\\ (\mathrm{adj}(B){)}^T=\mathrm{adj}\left(A\right)\\ \mathrm{adj}\left(B^T\right)=\mathrm{adj}\left(A\right)\end{array} \end{gathered}$$

$$\begin{gathered} \begin{array}{r}\mathrm{adj}\left(adj\left(B^T\right)\right)=\mathrm{adj}\left(adj\left(A\right)\right)\\ \Rightarrow \left|B^T\right|B^T=\left|A\right|A\end{array} \\ \begin{array}{c}{\left|B^T\right|}^2=|A{|}^2⟹\left|B^T\right|=\pm |A|\\ \\ \Rightarrow B^T=A\\ \begin{array}{r}\therefore 2A=\mathrm{adj}\left(A^T\right)\\ \left|2A\right|=\left|\mathrm{adj}\left(A^T\right)\right|\\ 8\left|A\right|={\left|A^T\right|}^2\\ \left|A\right|=\left|B\right|=8\end{array}\end{array} \end{gathered}$$

$$\begin{gathered} {\left(\det \mathrm{A}\right)}^2+{\left(\det \mathrm{B}\right)}^2=128 \\ \Rightarrow K=sum of digits=11 \\ \therefore \left[\frac{K}{2}\right]=5 \end{gathered}$$