Question

For any three complex numbers Z₁, Z₂, Z₃ if

D =$\left| \begin{matrix} 1 & Z_{1} & {\bar{Z}}_{1} \\ 1 & Z_{2} & {\bar{Z}}_{2} \\ 1 & Z_{3} & {\bar{Z}}_{3} \end{matrix} \right|$ ,

• A. The real part of D is always positive
• B. The real part of D is always negative
• C. The real part of D is always zero
• D. The imaginary part of D is always non- zero

Answer 1

Answer: (C)

The real part of D is always zero

Answer 2

Sol. D = $\left| \begin{matrix} 1 & x_{1} + iy_{1} & x_{1} - iy_{1} \\ 1 & x_{2} + iy_{2} & x_{2} - iy_{2} \\ 1 & x_{3} + iy_{3} & x_{3} - iy_{3} \end{matrix} \right|$= $\left| \begin{matrix} 1 & x_{1} & - iy_{1} \\ 1 & x_{2} & - iy_{2} \\ 1 & x_{3} & - iy_{3} \end{matrix} \right|$+

$\left| \begin{matrix} 1 & iy_{1} & x_{1} \\ 1 & iy_{2} & x_{2} \\ 1 & iy_{3} & x_{3} \end{matrix} \right|$ = –2i $\left| \begin{matrix} 1 & x_{1} & y_{1} \\ 1 & x_{2} & y_{2} \\ 1 & x_{3} & y_{3} \end{matrix} \right|$