Question

If A = $\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$, then $|A||AdjA|$ is equal to

• A. $a^{25}$
• B. $a^{27}$
• C. $a^{81}$
• D. none of these

Answer 1

The correct option is D none of these

We know $A\left(adjA\right)=|A|I$
$\Rightarrow |A||adjA|=|A{|}^n$
A = $\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$ i.e $n=3$
$|A|=\left|\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right|=a^3$
$\therefore |A||adjA|=(|A{|}^3{)}^3=A^9$
Hence, option D.