Question

If $A=\left[\begin{array}{cc}-4& -1\\ 3& 1\end{array}\right]$, then the determinant of the matrix $(A^{2016}-2A^{2015}-A^{2014})$ is :

• A. 2014
• B. -175
• C. 2016
• D. -25

Answer 1

Answer: (D)

$A=\left[\begin{array}{cc}-4& -1\\ 3& 1\end{array}\right]$
$A^2=\left[\begin{array}{cc}-4& -1\\ 3& 1\end{array}\right]\left[\begin{array}{cc}-4& -1\\ 3& 1\end{array}\right]=\left[\begin{array}{cc}13& 3\\ -9& -2\end{array}\right]$
$A^2-2A-I=\left[\begin{array}{cc}13& 3\\ -9& -2\end{array}\right]\left[\begin{array}{cc}-8& -2\\ 6& 2\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}20& 5\\ -15& -5\end{array}\right]$
And $|A|=-1$
$\Rightarrow \left|A^{2016}-2A^{2015}-A^{2014}\right|=|A{|}^{2014}$
$\left|A^2-2A-I\right|=1\left|\begin{array}{cc}20& 5\\ -15& -5\end{array}\right|=\left(-100+75\right)=-25$