Question:
If the matrices $\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3\end{array}\right], \mathrm{B}=\operatorname{adj} \mathrm{A}$
and $\mathrm{C}=3 \mathrm{~A}$, then $\frac{|\operatorname{adj} \mathrm{B}|}{|\mathrm{C}|}$ is equal to :
Correct Option: 1
Solution:
$|A|=\left|\begin{array}{ccc}1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3\end{array}\right|=((9+4)-1(3-4)+2(-1-3))$
$=13+1-8=6$
$|\operatorname{adj} B|=|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(n-1)^{2}}=|A|^{4}=(36)^{2}$
$|C|=|3 A|=3^{3} \times 6$
Hence, $\frac{|a d j B|}{|C|}=\frac{36 \times 36}{3^{3} \times 6}=8$
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