Solution:

Option(B)
To find: Value of $x$
We have, $\left|\begin{array}{ccc}3 x-8 & 3 & 3 \\ 3 & 3 x-8 & 3 \\ 3 & 3 & 3 x-8\end{array}\right|=0$
Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$
$\Rightarrow\left|\begin{array}{ccc}
3 x-11 & 11-3 x & 0 \\
3 & 3 x-8 & 3 \\
3 & 3 & 3 x-8
\end{array}\right|=0$
Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}$
$\Rightarrow\left|\begin{array}{ccc}
3 x-11 & 11-3 x & 0 \\
0 & 3 x-11 & 11-3 x \\
3 & 3 & 3 x-8
\end{array}\right|=0$
Expanding along $R_{1}$
$\begin{array}{l}
\Rightarrow(3 x-11)\{(3 x-11)(3 x-8)-(3)(11-3 x)\}-(11-3 x)\{(0)((3 x-8)-(11-3 x)(3)\}=0 \\
\Rightarrow(3 x-11)\{(3 x-11)(3 x-8+3)\}-(11-3 x)\{-(11-3 x)(3)\}=0 \\
\left.\Rightarrow(3 x-11)^{2}(3 x-5)\right\}+(3 x-11)\{(3 x-11)(3)\}=0 \\
\left.\left.\Rightarrow(3 x-11)^{2}(3 x-5)\right\}+(3 x-11)^{2}(3)\right\}=0 \\
\Rightarrow(3 x-11)^{2}(3 x-5+3)=0 \\
\Rightarrow(3 x-11)^{2}(3 x-2)=0 \\
\Rightarrow x=\frac{11}{3}, \text { Or, } x=\frac{2}{3}
\end{array}$