Let  $f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& \sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& \sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\sin 2x\end{array}\right|$, $x\in [\frac{\pi }{6},\frac{\pi }{3}]$ . If  $\alpha$ and  $\beta$  respectively are the maximum and the minimum values of  $f$, then

$C_1\to C_1+C_2+C_3$

$f\left(x\right)=\left|\begin{array}{ccc}2+\sin 2x& {\cos }^{2}x& \sin 2x\\ 2+\sin 2x& 1+{\cos }^{2}x& \sin 2x\\ 2+\sin 2x& {\cos }^{2}x& 1+\sin 2x\end{array}\right|$

$f\left(x\right)=(2+\sin 2x)\left|\begin{array}{ccc}1& {\cos }^{2}x& \sin 2x\\ 1& 1+{\cos }^{2}x& \sin 2x\\ 1& {\cos }^{2}x& 1+\sin 2x\end{array}\right|$

$R_2\to R_2-R_1$

$R_3\to R_3-R_1$

$f\left(x\right)=2+\sin 2x)\left|\begin{array}{ccc}1& {\cos }^{2}x& \sin 2x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$

$=(2+\sin 2x)\left(1\right)=2+\sin 2x$

$=\sin 2x\in [\frac{\sqrt{3}}{2},1]$

Hence $2+\sin 2x\in [2+\frac{\sqrt{3}}{2},3]$