Let be a positive real number and let A= bmatrix 2k-1&2 k &2 k 2 k &1&-2k -2 k &2k&-1…

MATHEMATICS · Subjective · CBSE · Mathematics · Determinants · Determinants

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(\operatorname{adj} A)+\det(\operatorname{adj} B)=10^6, then [k][k] is equal to ‾\underline{\hspace{1cm}}. Here, adj⁡M\operatorname{adj} M denotes the adjugate of a square matrix MM, and [k][k] denotes the greatest integer less than or equal to kk.

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