Question

Let $k$ be a positive real number and let
$A =\begin{bmatrix}2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1\end{bmatrix}$ and $B =\begin{bmatrix}0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0\end{bmatrix}$.
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

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