Let $k\in R^{+}A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right],B=\left[\begin{array}{ccc}0& 2k-1& \sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$ If $\left|adjA\right|+\left|adjB\right|={10}^6$ then $\left[k\right]=\_\_\_\_\_\_\_ [ . ] is G.I.F.$

Clearly B is skew symmetric matrix of order 3

$\therefore \left|B\right|=0\Rightarrow \left|adjB\right|=0$

$\left|adjA\right|={\left|A\right|}^2$

$\left|A\right|=(2k-1)\{-1+4k^2\}-2\sqrt{k}\left\{-2\sqrt{k}-2\sqrt{k}\left(2k\right)\right\}+2\sqrt{k}\{4k\sqrt{k}+2\sqrt{k}\}$

$=(4k^2-1)(2k-1)+4k+8k^2+8k^2+4k=8k^3-4k^2-2k+1+4k+8k^2+8k^2+4k$

$8k^3+12k^2+6k+1={(2k+1)}^3={10}^3$

$2k+1=10$

$k=\frac{9}{2}=4.5\Rightarrow \left[k\right]=4$