Question

If the matrices $A=\left[\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right],B=\text{adj}A$ and $C=3A$, then $\frac{|adjB|}{|C|}$ is equal to :

• A. $16$
• B. $2$
• C. $8$
• D. $72$

Answer 1

The correct option is C $8$

$A=\left[\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right]\Rightarrow |A|=\left|\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right|$
$=13+1-8=6$

$B=\text{adj}\left(A\right)$
$\Rightarrow |\text{adj}B|=|\text{adj}\left(\text{adj}A\right)|$
$\Rightarrow |A{|}^4=6^4$

$|C|=|3A|=3^3|A|=3^3\times 6$

$\frac{|\text{adj}B|}{|C|}=\frac{6^4}{3^3\times 6}=\frac{2^3\times 3^3}{3^3}=8$