Let k be a positive real number and let

$\begin{array}{r}A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right]\end{array},B=\left[\begin{array}{ccc}0& 2k-1& \sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$ $\text{ If }\det (\mathrm{Adj}(A\left)\right)+\det (\mathrm{Adj}(B\left)\right)={10}^6\text{ then }\left[k\right]\text{ is equal to }$

$\begin{array}{r}\det \left(A\right)=\left|\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right|\\ =\left|\begin{array}{ccc}2k-1& 0& 2\sqrt{k}\\ 2\sqrt{k}& 1+2k& -2k\\ -2\sqrt{k}& 2k+1& -1\end{array}\right|\end{array}$

$\begin{array}{l}\left(\mathrm{By}{C}_2\to C_2-C_3\right)\\ =\left|\begin{array}{ccc}2k-1& 0& 2\sqrt{k}\\ 4\sqrt{k}& 0& 1-2k\\ -2\sqrt{k}& 2k+1& -1\end{array}\right|\left(R_2\to R_2-R_1\right)\\ =(2k+1{)}^3\end{array}$

B is a skew- symmetric matrix of odd order therefore det (B)= 0

$\text{ Now det }(\mathrm{adj}(A\left)\right)+\det (\mathrm{adj}(B\left)\right)={10}^6$

$\begin{array}{l}\Rightarrow {\left\{(2k+1{)}^3\right\}}^2+0={10}^6\\ \Rightarrow 2k+1=0\Rightarrow k=4.5\Rightarrow \left[k\right]=4\end{array}$