Question

If $A=\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right],$ then the value of the determinant $|A^{2009}-5A^{2008}|$ is

• A. $-6$
• B. $-5$
• C. $-4$
• D. $4$
• E. $6$

Answer 1

Answer: (A)

Given, $A=\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]$
$\Rightarrow$ $|A|=5-6=-1$
$\therefore$ $|A^{2009}-5A^{2008}|=|A^{2008}||A-5I|$
$={(-1)}^{2008}\left|\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]-\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]\right|$
$=\left|\begin{array}{cc}-4& 2\\ 3& 0\end{array}\right|=-6$