Question

Let $A=\left[\begin{array}{cc}5& 6\\ 3& 4\end{array}\right]$ be a given matrix. The determinant value of $A^{2010}–5A^{2009}$ is

• A. $–{7.2}^{2009}$
• B. $(–11){.2}^{2011}$
• C. $2^{2011}$
• D. $2^{2010}(–9)$

Answer 1

The correct option is D $2^{2010}(–9)$

We have,
$A–5I=\left[\begin{array}{cc}0& 6\\ 3& -1\end{array}\right]$
$\therefore |A–5I|=–18$
$|A^{2010}-5A^{2009}|=|A^{2009}(A-5I)|$
$=|A{|}^{2009}|A-5I|$
$=2^{2009}\times -18=(-9){.2}^{2010}$