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Let $k$ be a positive real number and let $\begin{aligned} A & =\left[\begin{array}{ccc}2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1\end{array}\right] \text { and } \\ B & =\left[\begin{array}{ccc}0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0\end{array}\right]\end{aligned}$
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 $k$ ].
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 $k$ ].
Solution:
2768 Upvotes
Verified Answer
The correct answer is:
4
$|A|=(2 k+1)^3,|B|=0$
But $\operatorname{det}(\operatorname{adj} A)=\operatorname{det}(\operatorname{adj} B)=10^6$
$$
\begin{aligned}
& \Rightarrow(2 k+1)^6=10^6 \\
& \Rightarrow k=\frac{9}{2} \Rightarrow[k]=4
\end{aligned}
$$
But $\operatorname{det}(\operatorname{adj} A)=\operatorname{det}(\operatorname{adj} B)=10^6$
$$
\begin{aligned}
& \Rightarrow(2 k+1)^6=10^6 \\
& \Rightarrow k=\frac{9}{2} \Rightarrow[k]=4
\end{aligned}
$$
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