Question

If matrix A is given by $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right]$, then the determinant of $A^{2005}-6A^{2004}$ is

• A. $2^{2006}$
• B. $(-11)2^{2005}$
• C. $-2^{2005}$
• D. $(-9)2^{2004}$

Answer 1

Answer: (B)

$(-11)2^{2005}$

Given, $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right]$
$|A|=2$
Now, $|A^{2005}-6A^{2004}|=|A{|}^{2004}|A-6I|$
$=2^{2004}\left|\begin{array}{cc}0& 11\\ 2& -2\end{array}\right|$
$=2^{2004}(-22)$
$=2^{2005}(-11)$