Question

Let k be a positive real number and let $A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right]$
$B=\left[\begin{array}{ccc}0& 2k-1& \sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$
If det $\left(Adj\right(A\left)\right)+det\left(Adj\right(B\left)\right)={10}^o$ then $\left[k\right]$ is equal to.

• A. $4$
• B. $6$
• C. $0$
• D. $1$

Answer 1

Answer: (A)

$4$

det $\left(A\right)=\left|\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right|$

Apply $C_2\to C_2-C_3$

$=\left|\begin{array}{ccc}2k-1& 2\sqrt{k}-2\sqrt{2}& 2\sqrt{k}\\ 2\sqrt{k}& 1+2k& -2k\\ -2\sqrt{k}& 2k+1& -1\end{array}\right|$

$=\left|\begin{array}{ccc}2k-1& 0& 2\sqrt{k}\\ 2\sqrt{k}& 1+2k& -2k\\ -2\sqrt{k}& 2k+1& -1\end{array}\right|$

Apply $R_2\to R_2-R_3$

$=\left|\begin{array}{ccc}2k-1& 0& 2\sqrt{k}\\ 4\sqrt{k}& 0& 1-2k\\ -2\sqrt{k}& 2k+1& -1\end{array}\right|$

$=(2k+1{)}^3$

$\because$ B is a skew symmetric matrix of odd order therefore det $\left(B\right)=0$

Now det $\left(Adj\right(A\left)\right)+det\left(adj\right(B\left)\right)={10}^6$

$\Rightarrow \left\{\right(2k+1{)}^3{\}}^2+0={10}^6$

$\Rightarrow 2k+1=0$

$\Rightarrow k=4,5$

$\Rightarrow \left[k\right]=4$.