Question

Let K be a positive real number and $A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 2k-1& \sqrt{k}\\ 1-2k& 0& 2\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$. If $det\left(adjA\right)+det\left(adjB\right)={10}^6$, 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].

Answer 1

$det(adjA{)}^n=det(A{)}^{n-1}$....(Property)

$A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right]$

$det(A{)}^2+det(B{)}^2={10}^6$.....(1)

$det\left(A\right)=(2k-1)(4k^2-1)+4k(2k+1)+4k(2k+1)$

$det\left(A\right)=(2k+1{)}^3$.....(2)

Since Matrix $B$ is a skew symmetric matrix, its determinant will be $0$.

$det\left(B\right)=0$......(3)

From (1), (2) and (3), we get

$\Rightarrow (2k+1{)}^6={10}^6$

$\Rightarrow 2k+1=10$

$\Rightarrow k=4.5$

So, greatest integer of $4.5$ is $4$