Question

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

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

Answer 1

Answer: (A)

$-6$

Answer 2

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