Mathematics

Mathematics

Problem

Let kk be a positive real number and let A=[2k−12k2k2k1−2k−2k2k−1]andB=[02k−1k1−2k02k−k−2k0]A = \begin{bmatrix} 2k-1 & 2\sqrt{k} & 2\sqrt{k} \\ 2\sqrt{k} & 1 & -2k \\ -2\sqrt{k} & 2k & -1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 0 & 2k-1 & \sqrt{k} \\ 1-2k & 0 & 2\sqrt{k} \\ -\sqrt{k} & -2\sqrt{k} & 0 \end{bmatrix} If det⁡(adj A)+det⁡(adj B)=106\det(\text{adj } A) + \det(\text{adj } B) = 10^6 , then [k][k] is equal to [Note: adj M\text{adj } M denotes the adjoint of a square matrix MM and [k][k] denotes the largest less than or equal to kk ]

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