Question

The value of the determinant $\left|\begin{array}{ccc}\sqrt{6}& 2i& 3+\sqrt{6}\\ \sqrt{12}& \sqrt{3}+\sqrt{8}i& 3\sqrt{2}+\sqrt{6}i\\ \sqrt{18}& \sqrt{2}+\sqrt{12}i& \sqrt{27}+2i\end{array}\right|$ is

• A. complex
• B. real
• C. irrational
• D. rational

Answer 1

Answer: (A), (B)

The correct options are
A complex
B real

$\left|\begin{array}{ccc}\sqrt{6}& 2i& 3+\sqrt{6}\\ \sqrt{12}& \sqrt{3}+\sqrt{8}i& 3\sqrt{2}+\sqrt{6}i\\ \sqrt{18}& \sqrt{2}+\sqrt{12}i& \sqrt{27}+2i\end{array}\right|$

$R_2\to R_2-\sqrt{2}{R}_1$

$R_3\to R_3-\sqrt{3}{R}_1$

$\Rightarrow △=\sqrt{6}\left|\begin{array}{ccc}1& 2i& 3+\sqrt{6}\\ 0& \sqrt{3}& \sqrt{6}i-2\sqrt{3}\\ 0& \sqrt{2}& 2i-3\sqrt{2}\end{array}\right|$

$\Rightarrow \sqrt{6}\left|\begin{array}{cc}\sqrt{3}& \sqrt{6}i-2\sqrt{3}\\ \sqrt{2}& 2i-3\sqrt{2}\end{array}\right|$

$\Rightarrow △$ is real and complex.

A and B