Question

Matrix $A$ is given by $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right]$ then the determinant of $A^{2015}-6A^{2014}$ is.

• A. $2^{2016}$
• B. $(-11)2^{2015}$
• C. ${-2}^{2015}\times 7$
• D. $(-9)2^{2014}$

Answer 1

The correct option is B $(-11)2^{2015}$

$A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right]\left|A\right|=24-22=2$

Now, here are certain properties of determinants that will be useful in further calculations

$i\left)det\right(AB)=det(A)\times det(B\left)ii\right)det\left(A^n\right)={\left[det\right(A\left)\right]}^n$

So,

$\therefore det(A^{2015}-6A^{2014})=det\left(A^{2014}\right(A-6I\left)\right)=det\left(A^{2014}\right)det(A-6I)={\left(detA\right)}^{2014}det\left(\left[\begin{array}{cc}0& 11\\ 2& -2\end{array}\right]\right)=2^{2014}(-22)=2^{2015}(-11)$