If $A=\left [ \frac{5}{111} \frac{-3}{336}\right ]$ and det $(-3A^{2013}+A^{2014})=\alpha ^{\alpha }\beta ^{2}(1+\gamma +\gamma ^{2})$ then, where $\alpha ,\beta ,\gamma$ are integers

Let,

$\begin{array}{l} |A|=\left|\begin{array}{cc} 5 & -3 \\ 111 & 336 \end{array}\right| \\ |A|=2013 \end{array}$

$\begin{array}{l}\text { Now, } \operatorname{det}\left(-3 A^{2013}+A^{2014}\right) \\ =\left|A^{2013}(-31+A)\right| \\ =|A|^{2013}|A-31| \\ =(2013)^{2013} \cdot\left|\begin{array}{ll}2 & -3 \\ 111 & 333\end{array}\right| \\ =(2013)^{2013} \cdot(111 \times 9) \\ =(2013)^{2013} \cdot 3^{2}(1+10+100)\end{array}$

On comparing we get,

$\alpha=2013, \beta=3, \gamma=10$

$\therefore$ correct ans, $\gamma=0$