Let $k$ be a positive real number and let $\begin{aligned} A & =\left[\begin{array}{ccc}2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \ 2 \sqrt{k} & 1 & -2 k \ -2 \sqrt{k} & 2 k & -1\end{array} ight] ext { and } \ B & =\left[\begin{array}{ccc}0 & 2 k-1 & \sqrt{k} \ 1-2 k & 0 & 2 \sqrt{k} \ -\sqrt{k} &…

Question

Let $k$ be a positive real number and let $\begin{aligned} A & =\left[\begin{array}{ccc}2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \ 2 \sqrt{k} & 1 & -2 k \ -2 \sqrt{k} & 2 k & -1\end{array}ight] 	ext { and } \ B & =\left[\begin{array}{ccc}0 & 2 k-1 & \sqrt{k} \ 1-2 k & 0 & 2 \sqrt{k} \ -\sqrt{k} & -2 \sqrt{k} & 0\end{array}ight]\end{aligned}$ If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[k]$ is equal to [Note : adj $M$ denotes the adjoint of a square matrix $M$ and $[k]$ denotes the largest integer less than or equal to $k$ ].

Aicharya · Accepted Answer

$|A|=(2 k+1)^3,|B|=0$ But $\operatorname{det}(\operatorname{adj} A)=\operatorname{det}(\operatorname{adj} B)=10^6$ $ \begin{aligned} & \Rightarrow(2 k+1)^6=10^6 \ & \Rightarrow k=\frac{9}{2} \Rightarrow[k]=4 \end{aligned} $

Let $k$ be a positive real number and let $\begin{aligned} A & =\left[\begin{array}{ccc}2 k-1 & 2 \sqrt{k} &…

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