If $\left({\sin }^{-1}\mathrm{x}+{\sin }^{-1}\mathrm{w}\right)\left({\sin }^{-1}\mathrm{y}+{\sin }^{-1}\mathrm{z}\right)={\mathrm{\pi }}^2,\text{ then }\mathrm{D}=\left|\begin{array}{cc}{\mathrm{x}}^{{\mathrm{N}}_1}& {\mathrm{y}}^{{\mathrm{N}}_2}\\ {\mathrm{z}}^{{\mathrm{N}}_3}& {\mathrm{w}}^{{\mathrm{N}}_4}\end{array}\right|\left({\mathrm{N}}_1,{\mathrm{N}}_2,{\mathrm{N}}_3,{\mathrm{N}}_4\in \mathrm{N}\right)$

$\begin{array}{l}{\sin }^{-1}\mathrm{x}+{\sin }^{-1}\mathrm{w}={\sin }^{-1}\mathrm{y}+{\sin }^{-1}\mathrm{z}=-\mathrm{\pi }\\ \therefore \mathrm{x}=\mathrm{y}=\mathrm{z}=\mathrm{w}=1\text{ or }\mathrm{x}=\mathrm{y}=\mathrm{z}=\mathrm{w}=-1\end{array}$
Hence, the maximum value of $\left|\begin{array}{cc}{\mathrm{x}}^{{\mathrm{N}}_1}& {\mathrm{y}}^{{\mathrm{N}}_2}\\ {\mathrm{z}}^{{\mathrm{N}}_3}& {\mathrm{w}}^{{\mathrm{N}}_4}\end{array}\right|=\left|\begin{array}{cc}1& -1\\ 1& 1\end{array}\right|=2$ and minimum value $\left|\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right|=-2$
Also, there are 16 different determinants as each place value is either 1 or –1.