If f (x) = $\left|\begin{array}{ccc}1 + {\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 4\sin 2\mathrm{x}\\ {\sin }^2\mathrm{x}& 1+{\cos }^2\mathrm{x}& 4\sin 2\mathrm{x}\\ {\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 1+4\sin 2\mathrm{x}\end{array}\right|$ then the maximum of f(x) is

$\begin{array}{rcl}{\mathrm{R}}_2& \to & {\mathrm{R}}_2-{\mathrm{R}}_1, {\mathrm{R}}_3\Rightarrow {\mathrm{R}}_3-{\mathrm{R}}_1\\ \mathrm{f}\left(\mathrm{x}\right)& =& \left|\begin{array}{ccc}1+{\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 4\sin 2\mathrm{x}\\ -1& 1& 0\\ -1& 0& 1\end{array}\right|\\ & =& (1+{\sin }^2\mathrm{x})+{\cos }^2\mathrm{x}+4\sin 2\mathrm{x}\\ & =& 2+4\sin 2\mathrm{x}\end{array}$

$\therefore$ maximum value =6; $(-\le \sin 2\mathrm{x}\le 1)$