If A = - 4 - 1 3 1 , then the determinant of the matrix A 2016 - 2 A 2015 - A 2014 is :

$A=\left[\begin{array}{cc}-4& -1\\ 3& 1\end{array}\right]$

$\Rightarrow A^2=\left[\begin{array}{cc}-4& -1\\ 3& 1\end{array}\right] \left[\begin{array}{cc}-4& -1\\ 3& 1\end{array}\right]=\left[\begin{array}{cc}13& 3\\ -9& -2\end{array}\right]$

and $\left|A\right|=-1$

Now, $A^{2016}-2A^{2015}-A^{2014}=A^{2014} \left(A^2-2A-I\right)$

 $\Rightarrow \left|A^{2016}-2A^{2015}-A^{2014}\right|=\left|A^{2014}\right| \left|A^2-2A-I\right|$

$A^2-2A-I=\left[\begin{array}{cc}20& 5\\ -15& -5\end{array}\right]$

$={\left|A\right|}^{2014} \left|\begin{array}{cc}20& 5\\ -15& -5\end{array}\right|$

$=-25$