Question

The value of $k(k\in R)$ such that $\text{det}(\text{adj}(\text{adj}A))=16$ where $A=\left[\begin{array}{ccc}k& 1& 2\\ 0& -1& 1\\ 4& 1& 1\end{array}\right]$, is equal to

• A. 5,7
• B. 2,7
• C. 5,2
• D. 5,3

Answer 1

Answer: (A)

$|A|=\left|\begin{array}{ccc}k& 1& 2\\ 0& -1& 1\\ 4& 1& 1\end{array}\right|=k(-1-1)-1(0-4)+2(0+4)$
$=-2k+12=12-2k$
$|\text{adj}(\text{adj}A)|=|A{|}^4=(12-2k{)}^4=16$
$\Rightarrow 2k-12=\pm 2$
$\Rightarrow k=7,5$