Question

The value of determinant
$\left|\begin{array}{ccc}\sqrt{\left(13\right)}+\sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{\left(15\right)}+\sqrt{\left(26\right)}& 5& \sqrt{10}\\ 3+\sqrt{\left(65\right)}& \sqrt{\left(15\right)}& 5\end{array}\right|$ is:

• A. $5\sqrt{3}(5+\sqrt{6})$
• B. $-5(5-\sqrt{6})$
• C. $-5\sqrt{3}(5-\sqrt{3})$
• D. $-5\sqrt{3}(5-\sqrt{6})$

Answer 1

The correct option is D $-5\sqrt{3}(5-\sqrt{6})$

On taking $\sqrt{5}$ common from $C_2\text{and}{C}_3$, the given determinant can be written as:
$\Delta =(\sqrt{5}{)}^2\left|\begin{array}{ccc}\sqrt{\left(13\right)}+\sqrt{3}& 2& 1\\ \sqrt{\left(15\right)}+\sqrt{\left(26\right)}& \sqrt{5}& \sqrt{2}\\ 3+\sqrt{\left(65\right)}& \sqrt{3}& \sqrt{5}\end{array}\right|$
Now applying $C_1\to C_1-\sqrt{3}{C}_2-\sqrt{\left(13\right)}{C}_3$, we have
$\Delta =5\left|\begin{array}{ccc}-\sqrt{3}& 2& 1\\ 0& \sqrt{5}& \sqrt{2}\\ 0& \sqrt{3}& \sqrt{5}\end{array}\right|$
$\Rightarrow \Delta =-5\sqrt{3}(5-\sqrt{6})$