Question

If $z$ is a complex number $a_1,a_2,a_3,b_1,b_2,b_3$ are all real, then $\left|\begin{array}{ccc}{a}_{1}z+b_1\overline{z}& a_{2}z+b_2\overline{z}& a_{3}z+b_3\overline{z}\\ b_{1}z+a_1\overline{z}& b_{2}z+a_2\overline{z}& b_{3}z+a_3\overline{z}\\ b_{1}z+a_1& b_{2}z+a_2& b_{3}z+a_3\end{array}\right|=$

• A. $1$
• B. $2$
• C. $0$
• D. None of these

Answer 1

The correct option is C $0$

Using the product rule of detesminatr, we have

$\left|\begin{array}{ccc}{a}_{1}z+b_1\overline{z}& a_{2}z+b_2\overline{z}& a_{3}z+b_3\overline{z}\\ b_{1}z+a_1\overline{z}& b_{2}z+a_2\overline{z}& b_{3}z+a_3\overline{z}\\ b_{1}z+a_1& b_{2}z+a_2& b_{3}z+a_3\end{array}\right|=\left|\begin{array}{ccc}z& \overline{z}& 1\\ \overline{z}& z& 1\\ 1& z& z\end{array}\right|\left|\begin{array}{ccc}{a}_1& b_1& 0\\ a_2& b_2& 0\\ a_3& b_3& 0\end{array}\right|=0$