Find the K

Algebra Level 5

Let kk be a positive real number and let A=[2k12k2k2k12k2k2k1]A\quad =\quad \left[ \begin{matrix} 2k-1 & 2\sqrt { k } & 2\sqrt { k } \\ 2\sqrt { k } & 1 & -2k \\ -2\sqrt { k } & 2k & -1 \end{matrix} \right] and B=[02k1k12k02kk2k0].B\quad =\quad \left[ \begin{matrix} 0 & 2k-1 & \sqrt { k } \\ 1-2k & 0 & 2\sqrt { k } \\ -\sqrt { k } & -2\sqrt { k } & 0 \end{matrix} \right] . If det(adjA)+det(adjB)=106,\text{det}(adj\quad A)\quad +\quad \text{det}(adj\quad B)\quad =\quad { 10 }^{ 6 }, what is the value of [k]?[k]?

Note: [k][k] is the greatest integer less than or equal to k.k.

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