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 $(\text{adj}A)+\text{det}(\text{adj}B)=10^6$, then $[k]$ 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 $k]$.

Answer 1

Answer: 5

$|A|=(2k+1{)}^3,|B|=0$
(Since $B$ is a skew-symmetric matrix of order $3$ )
$\Rightarrow \text{det}(\text{adj}A)=|A{|}^{1-1}={\left((2k+1{)}^3\right)}^2=106$
$\Rightarrow 2k+1=10$
$\Rightarrow 2k=9$
$[k]=4.$