Determinants Matrices Question 92
Question: If x, y, z are complex numbers, and $ \Delta = \begin{vmatrix} 0 & -y & -z \\ {\bar{y}} & 0 & -x \\ {\bar{z}} & {\bar{x}} & 0 \\ \end{vmatrix} $ then $ \Delta $ is
Options:
A) Purely real
B) Purely imaginary
C) Complex
D) 0
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] We have $ \overline{\Delta }= \begin{vmatrix} 0 & -\overline{y} & -\overline{z} \\ y & 0 & -\overline{x} \\ z & x & 0 \\ \end{vmatrix}= \begin{vmatrix} 0 & y & z \\ -\overline{y} & 0 & x \\ -\overline{z} & -\overline{x} & 0 \\ \end{vmatrix} $ [Interchanging rows and columns] $ ={{(-1)}^{3}} \begin{vmatrix} 0 & -y & -z \\ \overline{y} & 0 & -x \\ \overline{z} & \overline{x} & 0 \\ \end{vmatrix} $ [Taking -1 common from each row] $ =-\Delta $
$ \therefore \overline{\Delta }+\Delta =0\Rightarrow 2Re(\Delta )=0 $
$ \therefore \Delta $ is purely imaginary.
BETA
