Question

If $A=\left|\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right|$, then the value of $\left|A\right|\left|adj\left(A\right)\right|$ is

• A. $a^3$
• B. $a^6$
• C. $a^9$
• D. $a^{27}$

Answer 1

Answer: (C)

$a^9$

$A=\left|\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right|=\Rightarrow |A|=a^3\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|=a^3\left|adj\right(A\left)\right|={\left|A\right|}^{n-1}$

Here $n=3$

$\Rightarrow \left|adj\right(A\left)\right|={\left|A\right|}^{3-1}={\left|A\right|}^2=a^6\Rightarrow |A|.\left|adj\right(A\left)\right|=a^3.a^6=a^9$

So option $C$ is correct