Question

If $\Delta = \left| \begin{matrix} \sqrt{6} & 2i & 3 + \sqrt{6} \\ \sqrt{12} & \sqrt{3} + \sqrt{8}i & 3\sqrt{2} + \sqrt{6}i \\ \sqrt{18} & \sqrt{2} + \sqrt{12}i & \sqrt{27} + 2i \end{matrix} \right|$, then ∆ is

• A. An integer
• B. A rational number
• C. An irrational number
• D. An imaginary number

Answer 1

Answer: (A)

An integer

Answer 2

Taking $\sqrt{6}$ common from C_1,

$R_{2} \rightarrow R_{2} - \sqrt{2}R_{1,}R_{3} \rightarrow R_{3} - \sqrt{3}R_{1}$

1 & 2i & 3 + \sqrt{6} \\ 0 & \sqrt{3} & \sqrt{6}i - 2\sqrt{3} \\ 0 & \sqrt{2} & 2i - 3\sqrt{2} \end{matrix} \right|$$ = $\left( C_{2} \rightarrow C_{2} - \sqrt{2}iC_{1} \right)$ = $\sqrt{6}( - 3\sqrt{6} + 2\sqrt{6}) = - 6$