If $A$ is a $3 \times 3$ matrix and $|A|=2$, then $|\operatorname{Adj}(\operatorname{Adj}(A))| =…
If $A$ is a $3 \times 3$ matrix and $|A|=2$, then $|\operatorname{Adj}(\operatorname{Adj}(A))| = |\operatorname{Adj}(A)|^{2}$.
\(32 \mathrm{~A}\)
\(64 \mathrm{~A}\)
\(16 \mathrm{~A}\)
\(8 \mathrm{~A}\)
Solution
As, $\operatorname{adj}(A)=|A|^{n-2} A$,
where $n$ is the order of matrix $A$.
and $|\operatorname{adj}(\operatorname{adj}(A))|=|A|^{(n-1)^2}$
So, $|\operatorname{adj}(\operatorname{adj}(A))|(\operatorname{adj}(\operatorname{adj}(A)))=|A|^{(n-1)^2}|A|^{n-2} A$
$=2^4 \cdot 2 A=2^5 A=32 A$.
$\because |A|=2$ and $n=3$
Hence, option (a) is correct.