Question

If matrix $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right],$ then the value of determinant of $A^{2015}-6A^{2014}$ is

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

Answer 1

The correct option is C $(-11)\times 2^{2015}$

$|A^{2015}-6A^{2014}|={\left|A\right|}^{2014}|A-6I|\cdots \left(i\right)$
Now, $A-6I=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right]-\left[\begin{array}{cc}6& 0\\ 0& 6\end{array}\right]=\left[\begin{array}{cc}0& 11\\ 2& -2\end{array}\right]$
Substituting in $\left(i\right),$ we get
$2^{2014}\left|\begin{array}{cc}0& 11\\ 2& -2\end{array}\right|=(-22)2^{2014}=-11(2{)}^{2015}$