Matrices and Determinants 3 Question 14
16. Let $k$ be a positive real number and let
$ A=\begin{bmatrix} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{bmatrix} \quad \text { and } $
$ B=\begin{bmatrix} 0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0 \end{bmatrix} $
If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^{6}$, then $[k]$ is equal to……
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Answer:
Correct Answer: 16. (4)
Solution:
- $|A|=(2 k+1)^{3},|B|=0$
But 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 \quad[k]=4 \end{aligned} $
