Question

Let $k$ be a positve 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 $B=\left[\begin{array}{ccc}-& 22k-1& 2\sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$. If $\text{det (adj A)+det(adj B)}={10}^6$, then $\left[k\right]$ is equal to

$[$Note: $\text{adj M}$ denotes the adjoint of a square matrix $M$ and $\left[k\right]$ denotes the largest integer less than or equal to $k]$

Answer 1

$|A|=(2k+1{)}^3,|B|=0$
$\Rightarrow \text{det (adj A)}\ne \text{det (adj B)}$ =(2k+1)^6 $={10}^6\Rightarrow k=\frac{9}{2}$
$\Rightarrow \left[k\right]=4$