Find the K

Algebra Level 5

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

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

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