Question

Without expanding, show that the value of the following determinant is zero:

\[\begin{vmatrix}\sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5\end{vmatrix}\]

Answer 1

\[\begin{vmatrix}\sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5\end{vmatrix}\]
\[ = \begin{vmatrix}\sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5\end{vmatrix} + \begin{vmatrix}\sqrt{23} & \sqrt{5} & \sqrt{5} \\ \sqrt{46} & 5 & \sqrt{10} \\ \sqrt{115} & \sqrt{15} & 5\end{vmatrix}\]
\[ = \sqrt{3}\begin{vmatrix}1 & \sqrt{5} & \sqrt{5} \\ \sqrt{5} & 5 & \sqrt{10} \\ \sqrt{3} & \sqrt{15} & 5\end{vmatrix} + \sqrt{23}\begin{vmatrix}1 & \sqrt{5} & \sqrt{5} \\ \sqrt{2} & 5 & \sqrt{10} \\ \sqrt{5} & \sqrt{15} & 5\end{vmatrix}\]
\[ = \sqrt{3} \times \sqrt{5}\begin{vmatrix}1 & 1 & \sqrt{5} \\ \sqrt{5} & \sqrt{5} & \sqrt{10} \\ \sqrt{3} & \sqrt{3} & 5\end{vmatrix} + \sqrt{23} \times \sqrt{5}\begin{vmatrix}1 & \sqrt{5} & 1 \\ \sqrt{2} & 5 & \sqrt{2} \\ \sqrt{5} & \sqrt{15} & \sqrt{5}\end{vmatrix}\]
\[ = 0 + 0\]
\[ = 0\]