Explanation

(a)
Given, $A = \left\lbrack \begin{array}{r}
1\ \ 2 \\
3\ \ 5
\end{array} \right\rbrack$
$\Rightarrow |A| = 5 - 6 = - 1$
$\therefore\left| A^{2009} - 5A^{2008} \right| = \left| A^{2008} \right||A - 5I|$
$= ( - 1)^{2008}\left| \left\lbrack \begin{array}{r}
1\ \ 2 \\
3\ \ 5
\end{array} \right\rbrack - \left\lbrack \begin{array}{r}
5\ \ 0 \\
0\ \ 5
\end{array} \right\rbrack \right|$
$= \left| \begin{array}{r}
- 4\ \ \ \ 2 \\
3\ \ \ \ \ 0
\end{array} \right| = - 6$