Question

Let k be a positive real number and let $A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 2k-1& \sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$ .

If det $\left(adjA\right)+det\left(adjB\right)={10}^6,then\left[k\right]$ is equal to
[Note: adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (B)

$4$

$|A|=(2k+1{)}^3,|B|=0$ (Since $B$ is a skew-symmetric matrix of order 3)
$\Rightarrow det\left(adjA\right)=|A{|}^{n-1}=\left(\right(2k+1{)}^3{)}^2={10}^6\Rightarrow 2k+1=10\Rightarrow 2k=9$
$\left[k\right]=4$.