Question

If $A=\left[\begin{array}{c}a\\ 0\\ 0\end{array}\begin{array}{c}0\\ a\\ 0\end{array}\begin{array}{c}0\\ 0\\ a\end{array}\right]$, then the value of $\left|A\right|\left|\text{Adj}A\right|$

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

Answer 1

Answer: (C)

$a^9$

Consider

$A=\left|\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right|$

Firstly we find the determinant of $A$ as shown below:

$\left|A\right|=a[\left(a\right)\left(a\right)-\left(0\right)\left(0\right)]-0[\left(0\right)\left(a\right)-\left(0\right)\left(0\right)]+0[\left(0\right)\left(0\right)-\left(0\right)\left(a\right)]$

$\left|A\right|=a\left[a^2\right]-0+0$

$\left|A\right|=a^3$

Now to find the adjoint of A, we calculate the cofactors of A, so let Cij be cofactor of aij in A. therefore the cofactors are as shown below:

$C_{11}=\left|\begin{array}{cc}a& 0\\ 0& a\end{array}\right|=a\left(a\right)-0\left(0\right)=a^2$

$\Rightarrow$$C_{11}=a^2$

$C_{12}=\left|\begin{array}{cc}0& 0\\ 0& a\end{array}\right|=0\left(a\right)-0\left(0\right)=0$

$\Rightarrow$$C_{12}=0$

$C_{13}=\left|\begin{array}{cc}0& a\\ 0& 0\end{array}\right|=0\left(0\right)-0\left(a\right)=0$

$\Rightarrow$$C_{13}=0$

$C_{21}=\left|\begin{array}{cc}0& 0\\ 0& a\end{array}\right|=0\left(a\right)-0\left(0\right)=0$

$\Rightarrow$$C_{21}=0$

$C_{22}=\left|\begin{array}{cc}a& 0\\ 0& a\end{array}\right|=a\left(a\right)-0\left(0\right)=a^2$

$\Rightarrow$$C_{22}=a^2$

$C_{23}=\left|\begin{array}{cc}a& 0\\ 0& 0\end{array}\right|=a\left(0\right)-0\left(0\right)=0$

$\Rightarrow$$C_{23}=0$

$C_{31}=\left|\begin{array}{cc}0& 0\\ a& 0\end{array}\right|=0\left(0\right)-a\left(0\right)=0$

$\Rightarrow$$C_{31}=0$

$C_{32}=\left|\begin{array}{cc}a& 0\\ 0& 0\end{array}\right|=a\left(0\right)-0\left(0\right)=0$

$\Rightarrow$$C_{32}=0$

$C_{33}=\left|\begin{array}{cc}a& 0\\ 0& a\end{array}\right|=a\left(a\right)-0\left(0\right)=a^2$

$\Rightarrow$$C_{33}=a^2$

Hence the \text{Adj}\,oint of $A$ is as follows:

$\text{Adj}A={\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]}^T={\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& a^2& 0\\ 0& 0& a^2\end{array}\right]}^T$

$\Rightarrow$$\text{Adj}A=\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& a^2& 0\\ 0& 0& a^2\end{array}\right]$

Now we find the deteminant of \text{Adj}\, A:

$\left|\text{Adj}A\right|=a^2[\left(a^2\right)\left(a^2\right)-\left(0\right)\left(0\right)]-0[\left(0\right)\left(a^2\right)-\left(0\right)\left(0\right)]+0[\left(0\right)\left(0\right)-\left(0\right)\left(a^2\right)]$

$\left|\text{Adj}A\right|=a^2\left[a^4\right]-0+0$

$\left|\text{Adj}A\right|=a^6$

Finally we find $\left|A\right||\text{Adj}.A|$

$\left|A\right||\text{Adj}.A|=a^2\left(a^6\right)$

$\left|A\right||\text{Adj}.A|=a^8$

Hence option $C$ is correct.