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

The correct option is D $-25$

$|A^{2016}-2A^{2015}-A^{2014}|$
$=|A{|}^{2014}|A^2-2A-I|$
$\text{Now},|A|=-4+3=-1$
$\text{and}{A}^2-2A-I=\left[\begin{array}{cc}20& 5\\ -15& -5\end{array}\right]$
$\Rightarrow |A^2-2A-I|=-25$
$\therefore |A^{2016}-2A^{2015}-A^{2014}|$
$=(-1{)}^{2014}\times (-25)=-25$