13+325515+265103+65155=

3255155103155+132552651015155=3×5121552335+3×5121252535C2→C2−C1=53111502305=−53[5−6]=518−253=152−253

13+325515+265103+65155=

3255155103155+132552651015155=3×5121552335+3×5121252535C2→C2−C1=53111502305=−53[5−6]=518−253=152−253

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$|\begin{array}{ccc} \sqrt{13} + \sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{26} & 5 & \sqrt{10} \\ 3 + \sqrt{65} & \sqrt{15} & 5 \end{array}| =$

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$15 \sqrt{2} - 25 \sqrt{3}$

$15 \sqrt{5} - 25 \sqrt{6}$

$25 \sqrt{2} + 15 \sqrt{3}$

answer is A.

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Detailed Solution

$\begin{array}{l} |\begin{array}{ccc} \sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5 \end{array}| + |\begin{array}{ccc} \sqrt{13} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{26} & 5 & \sqrt{10} \\ \sqrt{15} & \sqrt{15} & 5 \end{array}| \\ = \sqrt{3} \times 5 |\begin{array}{ccc} 1 & 2 & 1 \\ \sqrt{5} & \sqrt{5} & \sqrt{2} \\ \sqrt{3} & \sqrt{3} & \sqrt{5} \end{array}| + \sqrt{3} \times 5 |\begin{array}{ccc} 1 & 2 & 1 \\ \sqrt{2} & \sqrt{5} & \sqrt{2} \\ \sqrt{5} & \sqrt{3} & \sqrt{5} \end{array}| \\ C_{2} \to C_{2} - C_{1} \end{array}$
$\begin{array}{l} = 5 \sqrt{3} |\begin{array}{ccc} 1 & 1 & 1 \\ \sqrt{5} & 0 & \sqrt{2} \\ \sqrt{3} & 0 & \sqrt{5} \end{array}| \\ = - 5 \sqrt{3} [5 - \sqrt{6}] = 5 \sqrt{18} - 25 \sqrt{3} \\ = 15 \sqrt{2} - 25 \sqrt{3} \end{array}$